From pro- $$p$$ Iwahori–Hecke Modules to $$(\varphi,\Gamma)$$ -Modules, II

From pro- $$p$$ Iwahori–Hecke Modules to $$(\varphi,\Gamma)$$ -Modules, II Abstract Let $${\mathfrak o}$$ be the ring of integers in a finite extension field of $${\mathbb Q}_p$$, let $$k$$ be its residue field. Let $$G$$ be a split reductive group over $${\mathbb Q}_p$$, let $${\mathcal H}(G,I_0)$$ be its pro-$$p$$ Iwahori–Hecke $${\mathfrak o}$$-algebra. In [2] we introduced a general principle how to assign to a certain additionally chosen datum $$(C^{(\bullet)},\phi,\tau)$$ an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from finite length $${\mathcal H}(G,I_0)$$-modules to $$(\varphi^r,\Gamma)$$-modules. In this paper we concretely work out such data $$(C^{(\bullet)},\phi,\tau)$$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular $${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$$-modules with the set of $$(\varphi^r,\Gamma)$$-modules satisfying a certain symmetry condition. 1 Introduction Let $${\mathfrak o}$$ be the ring of integers in a finite extension field of $${\mathbb Q}_p$$, let $$k$$ be its residue field. Let $$G$$ be a split reductive group over $${\mathbb Q}_p$$, let $$T$$ be a maximal split torus in $$G$$, let $$I_0$$ be a pro-$$p$$-Iwahori subgroup fixing a chamber $$C$$ in the $$T$$-stable apartment of the semisimple Bruhat Tits building of $$G$$. Let $${\mathcal H}(G,I_0)$$ be the pro-$$p$$-Iwahori Hecke $${\mathfrak o}$$-algebra. Let $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ denote the category of $${\mathcal H}(G,I_0)$$-modules of finite $${\mathfrak o}$$-length. From a certain additional datum $$(C^{(\bullet)},\phi,\tau)$$ we constructed in [2] an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of étale $$(\varphi^r,\Gamma)$$-modules (with $$r\in{\mathbb N}$$ depending on $$\phi$$). For $$G={\rm GL}_2({\mathbb Q}_p)$$, when precomposed with the functor of taking $$I_0$$-invariants, this yields the functor from smooth $${\mathfrak o}$$-torsion representations of $${\rm GL}_2({\mathbb Q}_p)$$ (or at least from those generated by their $$I_0$$-invariants) to étale $$(\varphi,\Gamma)$$-modules which plays a crucial role in Colmez’ construction of a $$p$$-adic local Langlands correspondence for $${\rm GL}_2({\mathbb Q}_p)$$. In [2] we studied in detail the functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ when $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ for $$d\ge1$$. In [6] the situation has been analyzed for $$G={\rm SL}_{d+1}({\mathbb Q}_p)$$. The purpose of this paper is to explain how the general construction of [2] can be installed concretely for other classical matrix groups $$G$$ (as well as for $$G$$’s of type $$E_6$$, $$E_7$$). Recall that $$C^{(\bullet)}=(C=C^{(0)}, C^{(1)},C^{(2)},\ldots)$$ is a minimal gallery, starting at $$C$$, in the $$T$$-stable apartment, that $$\phi\in N(T)$$ is a “period” of $$C^{(\bullet)}$$ and that $$\tau$$ is a homomorphism from $${\mathbb Z}_p^{\times}$$ to $$T$$, compatible with $$\phi$$ in a suitable sense. The above $$r\in{\mathbb N}$$ is just the length of $$\phi$$. It turns out that $$\tau$$ must be a minuscule fundamental coweight (at least if the underlying root system is simple). Conversely, any minuscule fundamental coweight $$\tau$$ can be included into a datum $$(C^{(\bullet)},\phi,\tau)$$, in such a way that some power of $$\tau$$ is a power of $$\phi$$. For $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ we gave explicit choices of $$(C^{(\bullet)},\phi,\tau)$$ with $$r=1$$ in [2] (there are essentially just two choices, and these are “dual” to each other). In this paper we work out “privileged” choices $$(C^{(\bullet)},\phi,\tau)$$ for the classical matrix groups, as well as for $$G$$’s of type $$E_6$$, $$E_7$$. We mostly consider $$G$$ with connected center $$Z$$. Our choices of $$(C^{(\bullet)},\phi,\tau)$$ are such that $$\phi\in N(T)$$ projects (modulo $$ZT_0$$, where $$T_0$$ denotes the maximal bounded subgroup of $$T$$) to the affine Weyl group (viewed as a subgroup of $$N(T)/ZT_0$$). In particular, up to modifications by elements of $$Z$$ these $$\phi$$ can also be included into data $$(C^{(\bullet)},\phi,\tau)$$ for the other $$G$$’s with the same underlying root system, not necessarily with connected center. We indicate these modifications along the way. Notice that the $$\phi\in N(T)$$ considered in [2] for $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ does not project to the affine Weyl group, only its $$(d+1)$$st power (which is considered here) does so. But since the discussion is essentially the same, our treatment of the case $$A$$ here is very brief. In either case we work out the behavior of the functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ on those $${\mathcal H}(G,I_0)_k={\mathcal H}(G,I_0)\otimes_{\mathfrak o}k$$-modules which we call “standard supersingular”. Roughly speaking, these are induced from characters of the pro-$$p$$-Iwahori Hecke algebra of the corresponding simply connected group. Each irreducible supersingular $${\mathcal H}(G,I_0)_k$$-module is contained in (and in “most” cases is equal to) a standard supersingular $${\mathcal H}(G,I_0)_k$$-module (and the very few standard supersingular $${\mathcal H}(G,I_0)_k$$-modules which are not irreducible supersingular are easily identified). We show that our functor induces a bijection between the set of isomorphism classes (respectively, certain packets of such if $$G={\rm SO}_{2d+1}({\mathbb Q}_p)$$) of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and the set of (isomorphism classes of) étale $$(\varphi^r,\Gamma)$$-modules over $$k_{\mathcal E}=k((t))$$ which satisfy a certain symmetry condition (depending on the root system underlying $$G$$). They are direct sums of one-dimensional étale $$(\varphi^r,\Gamma)$$-modules, their dimension is the $$k$$-dimension of the corresponding standard supersingular $${\mathcal H}(G,I_0)_k$$-module. The interest in étale $$(\varphi^r,\Gamma)$$-modules lies in their relation with $${\rm Gal}_{{\mathbb Q}_p}$$-representations. For any $$r\in{\mathbb N}$$ there is an exact functor from the category of étale $$(\varphi^r,\Gamma)$$-modules to the category of étale $$(\varphi,\Gamma)$$-modules (it multiplies the rank by the factor $$r$$), and by means of Fontaine’s functor, the latter one is equivalent with the category of $${\rm Gal}_{{\mathbb Q}_p}$$-representations. In [2] we also explained that a datum $$(C^{(\bullet)},\phi)$$ alone, that is, without a $$\tau$$ as above, can be used to define an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of étale $$(\varphi^r,\Gamma_0)$$-modules, where $$\Gamma_0$$ denotes the maximal pro-$$p$$-subgroup of $$\Gamma\cong{\mathbb Z}_p^{\times}$$. Such data $$(C^{(\bullet)},\phi)$$ are not tied to co minuscule coweights and exist in abundance. We do not discuss them here. We hope that, beyond its immediate purposes as described above, this paper may also be a useful reference for explicit descriptions of the pro-$$p$$-Iwahori algebra (in particular with respect to the various Weyl groups involved) of classical matrix groups other than $${\rm GL}_d$$ (we could not find such descriptions in the literature). The outline is as follows. In Section 2 we explain the functor from étale $$(\varphi^r,\Gamma)$$-modules to étale $$(\varphi,\Gamma)$$-modules, and we introduce the “symmetric” étale $$(\varphi^r,\Gamma)$$-modules mentioned above, for each of the root systems $$C$$, $$B$$, $$D$$, and $$A$$. In Section 3, Lemma 3.1, we discuss the relation between the data $$(C^{(\bullet)},\phi,\tau)$$ and minuscule fundamental coweights. Our discussions of classical matrix groups $$G$$ in Section 4 are just concrete incarnations of Lemma 3.1, although in neither of these cases there is a need to make formal reference to Lemma 3.1. On the other hand, in our discussion of the cases $$E_6$$ and $$E_7$$ in Section 5 we do invoke Lemma 3.1. We tried to synchronize our discussions of the various matrix groups. As a consequence, arguments repeat themselves, and we do not write them out again and again. In the Appendix we record calculations relevant for the cases $$E_6$$ and $$E_7$$, carried out with the help of the computer algebra system sage. 2 $$\boldsymbol{(\varphi^r,\Gamma)}$$-modules We often regard elements of $${\mathbb F}_p^{\times}$$ as elements of $${\mathbb Z}_p^{\times}$$ by means of the Teichmüller lifting. In $${\rm GL}_2({\mathbb Z}_p)$$ we define the subgroups   Γ=(Zp×001),Γ0=(1+pZp001),N0=(1Zp01) and the elements   φ=(p001),ν=(1101),h(x)=(x00x−1),γ(x)=(x001), where $$x\in {\mathbb Z}_p^{\times}$$. Let $${\mathcal O}_{\mathcal E}^+={\mathfrak o}[[{\mathfrak N}_0]]$$ denote the completed group ring of $${\mathfrak N}_0$$ over $${\mathfrak o}$$. Let $${\mathcal O}_{\mathcal E}$$ denote the $$p$$-adic completion of the localization of $${{{\mathcal O}_{\mathcal E}^+}}$$ with respect to the complement of $$\pi_K{{{\mathcal O}_{\mathcal E}^+}}$$, where $$\pi_K\in{\mathfrak o}$$ is a uniformizer. In the completed group ring $$k_{\mathcal E}^+=k[[{\mathfrak N}_0]]$$ we put $$t=[\nu]-1$$. Let $$k_{\mathcal E}={\rm Frac}(k_{\mathcal E}^+)={{{\mathcal O}_{\mathcal E}}}\otimes_{{\mathfrak o}}k$$. We identify $$k_{\mathcal E}^+=k[[t]]$$ and $$k_{\mathcal E}=k((t))$$. For definitions and notational conventions concerning étale $$\varphi^r$$-modules and étale $$(\varphi^r,\Gamma)$$-modules we refer to [2]. Let $$r\in{\mathbb N}$$. Let $${\bf D}=({\bf D},{\varphi}^r_{{\bf D}})$$ be an étale $$\varphi^r$$-module over $${\mathcal O}_{\mathcal E}$$. For $$0\le i\le r-1$$ let $${\bf D}^{(i)}={\bf D}$$ be a copy of $${\bf D}$$. For $$1\le i\le r-1$$ define $$\varphi_{\widetilde{\bf D}}:{\bf D}^{(i)}\to{\bf D}^{(i-1)}$$ to be the identity map on $${\bf D}$$, and define $$\varphi_{\widetilde{\bf D}}:{\bf D}^{(0)}\to{\bf D}^{(r-1)}$$ to be the structure map $${\varphi}^r_{{\bf D}}$$ on $${\bf D}$$. Together we obtain a $${\mathbb Z}_p$$-linear endomorphism $$\varphi_{\widetilde{\bf D}}$$ on   D~=⨁i=0r−1D(i). Define an $${\mathcal O}_{\mathcal E}$$-action on $$\widetilde{\bf D}$$ by the formula $$x\cdot((d_i)_{0\le i\le r-1})=(\varphi^i_{{\mathcal O}_{\mathcal E}}(x)d_i)_{0\le i\le r-1}$$. Then the endomorphism $$\varphi_{\widetilde{\bf D}}$$ of $$\widetilde{\bf D}$$ is semilinear with respect to this $${\mathcal O}_{\mathcal E}$$-action, hence it defines on $$\widetilde{\bf D}$$ the structure of an étale $$\varphi$$-module over $${\mathcal O}_{\mathcal E}$$, see [2], Section 6.3. Let $$\Gamma'$$ be an open subgroup of $$\Gamma$$, let $${\bf D}$$ be an étale $$(\varphi^r,\Gamma')$$-module over $${\mathcal O}_{\mathcal E}$$. Define an action of $$\Gamma'$$ on $$\widetilde{\bf D}$$ by   γ⋅((di)0≤i≤r−1)=(γ⋅di)0≤i≤r−1. Lemma 2.1. The $$\Gamma'$$-action on $$\widetilde{\bf D}$$ commutes with $$\varphi_{\widetilde{\bf D}}$$ and is semilinear with respect to the $${\mathcal O}_{\mathcal E}$$-action, hence we obtain on $$\widetilde{\bf D}$$ the structure of an étale $$(\varphi,\Gamma')$$-module over $${\mathcal O}_{\mathcal E}$$. We thus obtain an exact functor from the category of étale $$(\varphi^r,\Gamma')$$-modules to the category of étale $$(\varphi,\Gamma')$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Proof. This is immediate from the respective properties of the $$\Gamma'$$-action on $${\bf D}$$. ■ The conjugation action of $$\Gamma$$ on $${\mathfrak N}_0$$ allows us to define for each $$x\in {\mathbb Z}_p^{\times}$$ an automorphism $$f\mapsto \gamma(x)f\gamma(x^{-1})$$ of $$k[[t]]=k_{\mathcal E}^+$$. Lemma 2.2. For $$x\in {\mathbb F}_p^{\times}$$ and $$m, n\in{\mathbb Z}_{\ge0}$$ we have   γ(x)tnprmγ(x−1)−(xt)nprm∈t(n+1)prmk[[t]]. (1) □ Proof. Denote again by $$x$$ be the representative of $$x$$ in $$[1,p-1]$$. We compute $$[\nu]^x-1=\sum_{j=1}^x{x\choose j}([\nu]-1)^x=\sum_{j=1}^x{x\choose j}t^j$$, hence   γ(x)tγ(x−1)−xt=([ν]x−1)−xt∈t2k[[t]],γ(x)tnγ(x−1)−(xt)n∈tn+1k[[t]],◼γ(x)tnprmγ(x−1)−(xt)nprm=(γ(x)tnγ(x−1)−(xt)n)prm∈t(n+1)prmk[[t]]. ■ Lemma 2.3. (a) Let $${\bf D}$$ be a one-dimensional étale $$(\varphi^r,\Gamma)$$-module over $$k_{{\mathcal E}}$$. There exists a basis element $$g$$ for $${\bf D}$$, uniquely determined integers $$0\le s({\bf D})\le p-2$$ and $$1\le n({\bf D})\le p^r-1$$ and a uniquely determined scalar $$\xi({\bf D})\in k^{\times}$$ such that   φDrg=ξ(D)tn(D)+1−prgγ(x)g−xs(D)g∈t⋅kE+⋅g for all $$x\in{\mathbb Z}_p^{\times}$$. Thus, one may define $$0\le k_{i}({\bf D})\le p-1$$ by $$n({\bf D})=\sum_{i=0}^{r-1}k_{i}({\bf D})p^i$$. One has $$n({\bf D})\equiv 0$$ modulo $$(p-1)$$. (b) For any given integers $$0\le s\le p-2$$ and $$1\le n\le p^r-1$$ with $$n\equiv 0$$ modulo $$(p-1)$$ and any scalar $$\xi\in k^{\times}$$ there is a uniquely determined (up to isomorphism) one-dimensional étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ with $$s=s({\bf D})$$ and $$n=n({\bf D})$$ and $$\xi=\xi({\bf D})$$. □ Proof. (a) Begin with an arbitrary basis element $$g_0$$ for $${\bf D}$$; then $$\varphi_{{\bf D}}^rg_0=F g_0$$ for some unit $$F\in k((t))=k_{{\mathcal E}}$$. After multiplying $$g_0$$ with a suitable power of $$t$$ we may assume $$F=\xi t^{m}(1+t^{n_0}F_0)$$ for some $$0\ge m\ge 2-p^r$$, some $$\xi\in k^{\times}$$, some $$n_0>0$$, and some $$F_0\in k[[t]]$$ (use $$t^{p^r}\varphi^r=\varphi^rt$$). For $$g_1=(1+t^{n_0}F_0)g_0$$ we then get $$\varphi_{{\bf D}}^rg_1=\xi t^{m}(1+t^{n_1}F_1)g_1$$ for some $$n_1>n_0>0$$ and some $$F_1\in k[[t]]$$. We put $$g_2=(1+t^{n_1}F_1)g_1$$. We may continue in this way. In the limit we get, by completeness, $$g=g_{\infty}\in{\bf D}$$ such that $$\varphi_{{\bf D}}^rg=\xi t^{m}g$$. It is clear that $$\xi({\bf D})=\xi$$ and $$n({\bf D})=p^{r}-1+m$$ are well-defined. Next, as $$\gamma(x)^{p-1}$$ (for $$x\in{\mathbb Z}_p^{\times}$$) is topologically nilpotent (in the topological group $$\Gamma$$) and acts by an automorphism on $${\bf D}$$, there is some unit $$F_x\in k[[t]]$$ with $$\gamma(x)g=F_xg$$. Thus, the action of $$\Gamma$$ on $${\bf D}$$ induces an action of $$\Gamma$$ on $$k[[t]]/tk[[t]]\cong k$$, and this action is given by a homomorphism $${\mathbb Z}_p^{\times}\cong\Gamma\to k^{\times}$$, $$x\mapsto x^{s({\bf D})}$$ for some $$0\le s({\bf D})\le p-2$$. Computing modulo $$t\cdot k^+_{{\mathcal E}}\cdot g$$ we have (with $$n=n({\bf D})$$, $$s=s({\bf D})$$, $$\xi=\xi({\bf D})$$)   xsg≡γ(x)g=ξ−1γ(x)tpr−n−1φDrg=ξ−1(γ(x)tpr−n−1γ(x−1))γ(x)φDrg≡(i)ξ−1(xt)pr−n−1γ(x)φDrg≡xsξ−1(xt)pr−n−1φDrg=xs+pr−n−1g. Here $$(i)$$ follows from $$\gamma(x)t^{p^r-n-1}\gamma(x^{-1})-(xt)^{p^r-n-1}\in t^{p^r-n}k[[t]]$$, formula (1). In short, we have $$g\equiv x^{p^r-n-1}g$$ for all $$x\in{\mathbb F}_p^{\times}$$. We deduce $$n\equiv0$$ modulo $$p-1$$. (b) Put $$D=k[[t]]=k^+_{\mathcal E}$$ and $$D^*={\rm Hom}_k^{\rm ct}(k^+_{\mathcal E},k)$$. Endow $$D^*$$ with an action of $$k^+_{\mathcal E}$$ by putting $$(\alpha\cdot\ell)(x)=\ell(\alpha\cdot x)$$ for $$\alpha\in k^+_{\mathcal E}$$, $$\ell\in D^*$$ and $$x\in D$$. For $$j\ge0$$ define $$\ell_j\in D^*$$ by   ℓj(∑i≥0aiti)=aj. Then $$\{\ell_j\}_{j\ge0}$$ is a $$k$$-basis of $$D^*$$, and we have $$t\ell_{j+1}=\ell_j$$ and $$t\ell_0=0$$. Define $$\varphi^r:D^*\to D^*$$ to be the $$k$$-linear map with $$\varphi^r(\ell_j)=\xi^{-1}\ell_{p^rj+n}$$. We then find   (tpr(φr(ℓj)))(∑i≥0aiti)=(ξ−1ℓprj+n)(∑i≥0aiti+pr)=ξ−1apr(j−1)+n,(φr(tℓj))(∑i≥0aiti)=ξ−1ℓpr(j−1)+n(∑i≥0aiti)=ξ−1apr(j−1)+n, (with $$a_i=0$$, respectively, $$\ell_i=0$$ for $$i<0$$). As $$t^{p^r}\varphi^r=\varphi^rt$$ in $$k^+_{\mathcal E}[\varphi^r]$$ this means that we have defined an action of $$k^+_{\mathcal E}[\varphi^r]$$ on $$D^*$$. Next, for $$x\in {\mathbb F}_p^{\times}$$ and $$b,m\in{\mathbb Z}_{\ge0}$$ we put   γ(x)⋅ℓ−b+n∑i=0m−1pri=x−s(γ(x)tbγ(x−1))ℓ−b+n∑i=0m−1pri, or equivalently,   γ(x)⋅tb(φr)mℓ0=x−s(γ(x)tbγ(x−1))(φr)mℓ0. (2) We claim that this defines an action of $$\gamma$$ on $$D^*$$. In order to check well definedness, that is, independence on the choice of $$b$$ and $$m$$ when only $$-b+n\sum_{i=0}^{m-1}p^{ri}$$ is given, it is enough to consider the equations $$t^b(\varphi^r)^m\ell_0=\xi t^{b+np^{rm}}(\varphi^r)^{m+1}\ell_0$$ and to compare the respective effect of $$\gamma(x)$$ (through formula (2)) on either side. Namely, we need to see   ξ−1x−s(γ(x)tbγ(x−1))(φr)mℓ0=x−s(γ(x)tb+nprmγ(x−1))(φr)m+1ℓ0, or equivalently,   (γ(x)tbγ(x−1))(ξ−1−(γ(x)tnprmγ(x−1))φr)(φr)mℓ0=0. For this it is enough to see that $$\xi^{-1}-(\gamma(x)t^{np^{rm}}\gamma(x^{-1})\varphi^r)$$ annihilates $$(\varphi^r)^m\ell_0$$. Now $$n<p^r$$ implies $$(n+1)p^{rm}> n\sum_{i=0}^mp^{ri}$$ and hence that $$t^{(n+1)p^{rm}}$$ annihilates $$(\varphi^r)^{m+1}\ell_0=\xi^{-m-1}\ell_{n\sum_{i=0}^mp^{ri}}$$. Therefore it is enough, by formula (1), to show that $$\xi^{-1}-(xt)^{np^{rm}}\varphi^r$$ annihilates $$(\varphi^r)^m\ell_0$$. But $$n\equiv 0$$ modulo $$(p-1)$$ means $$x^{np^{rm}}=1$$. Or claim follows since $$\xi^{-1}(\varphi^r)^m\ell_0=t^{np^{rm}}(\varphi^r)^{m+1}\ell_0$$. In fact, we have defined an action of $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$ on $$D^*$$. Namely, it is clear from the definitions that the relations $$\gamma(x)\cdot t= (\gamma(x)t\gamma(x^{-1}))\cdot \gamma(x)$$ in $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$ are respected. That the actions of $$\gamma(x)$$ and $$\varphi^r$$ on $$D^*$$ commute follows easily from the relations $$\gamma(x)\varphi^r=\varphi^r\gamma(x)$$ and $$t^{p^r}\varphi^r=\varphi^rt$$ in $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$. Notice that   γ(x)ℓ0=x−sℓ0 for all x∈Zp×. Now passing to the dual $$D\cong (D^*)^*$$ of $$D^*$$ yields a non-degenerate $$(\psi^r,\Gamma)$$-module over $$k^+_{\mathcal E}$$ with an associated étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ with $$s=s({\bf D})$$ and $$n=n({\bf D})$$ and $$\xi=\xi({\bf D})$$; this is explained in [2] Lemma 6.4. This dualization argument also proves the uniqueness of $${\bf D}$$. ■ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$C$$-symmetric if it admits a direct sum decomposition $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_1$$, $${\bf D}_2$$ satisfying the following conditions (1), (2C), and (3C): (1) $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$0\le i\le r-1$$. (2C) $$\xi({\bf D}_1)=\xi({\bf D}_2)$$. (3C) $$s({\bf D}_2)-s({\bf D}_1)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_1)$$ modulo $$(p-1)$$. (4) $$k_{\bullet}({\bf D}_1)\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$B$$-symmetric if $$r$$ is odd and if $${\bf D}$$ admits a direct sum decomposition $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_1$$, $${\bf D}_2$$ satisfying the following conditions (1), (2B), and (3B): (1) $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$0\le i\le r-1$$, and $$k_{\frac{r-1}{2}}({\bf D}_1)=k_{\frac{r-1}{2}}({\bf D}_2)$$ is even. (2B) For both $${\bf D}={\bf D}_1$$ and $${\bf D}={\bf D}_2$$ we have $$\xi({\bf D})=\prod_{i=0}^{r-1}(k_i({\bf D})!)^{-1}$$ and $$k_i({\bf D})=k_{r-1-i}({\bf D})$$ for all $$1\le i\le \frac{r-1}{2}$$. (3B) $$s({\bf D}_2)-s({\bf D}_1)\equiv k_{0}({\bf D}_1)-k_{r-1}({\bf D}_1)$$ modulo $$(p-1)$$. (4) $$k_{\bullet}({\bf D}_1)\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Lemma 2.4. The conjunction of the conditions (1), (2C), and (3C) (respectively, (1), (2B), and (3B)) is symmetric in $${\bf D}_1$$ and $${\bf D}_2$$. □ Proof. That each one of the conditions (1), (2C), and (2B) is symmetric even individually is obvious. Now $$n({\bf D})\equiv0$$ modulo $$p-1$$ implies $$\sum_{i=0}^{r-1}k_{i}({\bf D}_1)\equiv0$$ modulo $$p-1$$. Therefore $$s({\bf D}_2)-s({\bf D}_1)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_1)$$ and $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$i$$ (condition (1)) together imply $$s({\bf D}_1)-s({\bf D}_2)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_2)$$. Thus condition (3C) is symmetric, assuming condition (1). Similarly, condition (3B) is symmetric, assuming condition (1). ■ Definition. (i) Let $$\widetilde{\mathfrak S}_C(r)$$ denote the set of triples $$(n,s,\xi)$$ with integers $$1\le n\le p^r-2$$ and $$0\le s\le p-2$$ and scalars $$\xi\in k^{\times}$$ such that $$n\equiv 0$$ modulo $$(p-1)$$. Let $${\mathfrak S}_C(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_C(r)$$ by the involution   (∑i=0r−1kipi,s,ξ)↦(∑i=0r−1kr−i−1pi,s+∑i=0r−1iki,ξ). (Here and in the following, in the second component we mean the representative modulo $$p-1$$ belonging to $$[0,p-2]$$.) (ii) Let $$r$$ be odd and let $$\widetilde{\mathfrak S}_B(r)$$ denote the set of pairs $$(n,s)$$ with integers $$1\le n=\sum_{i=0}^{r-1}k_ip^i\le p^r-2$$ and $$0\le s\le p-2$$ such that $$n\equiv 0$$ modulo $$(p-1)$$ and such that $$k_i=k_{r-1-i}$$ for all $$1\le i\le \frac{r-1}{2}$$. Let $${\mathfrak S}_B(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_B(r)$$ by the involution   (∑i=0r−1kipi,s)↦(∑i=0r−1kr−i−1pi,s+k0−kr−1). □ Lemma 2.5. (i) Sending $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ to $$(n({\bf D}_1), s({\bf D}_1),\xi({\bf D}_1))$$ induces a bijection between the set of isomorphism classes of $$C$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_C(r)$$. (ii) Sending $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ to $$(n({\bf D}_1), s({\bf D}_1))$$ induces a bijection between the set of isomorphism classes of $$B$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_B(r)$$. □ Proof. This follows from Lemma 2.3. ■ Definition. Let $$r$$ be even. Let $$\widetilde{\mathfrak S}_D(r)$$ denote the set of triples $$(n,s,\xi)$$ with integers $$1\le n=\sum_{i=0}^{r-1}k_ip^i\le p^r-2$$ and $$0\le s\le p-2$$ and scalars $$\xi\in k^{\times}$$ such that $$n\equiv 0$$ modulo $$(p-1)$$ and such that $$k_i=k_{i+\frac{r}{2}}$$ for all $$1\le i\le \frac{r}{2}-2$$. We consider the following permutations $$\iota_0$$ and $$\iota_1$$ of $$\widetilde{\mathfrak S}_D(r)$$. The value of $$\iota_0$$ at $$(\sum_{i=0}^{r-1}k_ip^i,s,\xi)$$ is   (kr2+∑i=1r2−2kipi+kr−1pr2−1+k0pr2+∑i=r2+1r−2kipi+kr2−1pr−1,s+∑i=0r2−1ki,ξ). The value of $$\iota_1$$ at $$(\sum_{i=0}^{r-1}k_ip^i,s,\xi)$$ is   (∑i=0r−1kr−i−1pi,s+r−24(kr2+k0)+∑i=2r2−1(i−1)kr2−i,ξ) if $$\frac{r}{2}$$ is odd, whereas if $$\frac{r}{2}$$ is even the value is   (kr2−1+∑i=1r2−2kr−i−1pi+kr2pr2−1+kr−1pr2+∑i=r2+1r−2kr−i−1pi+k0pr−1,s+(r4−1)kr2+r4k0+∑i=2r2−1(i−1)kr2−ipi,ξ). It is straightforward to check that $$\iota_0^2={\rm id}$$ and $$\iota_0\iota_1=\iota_1\iota_0$$, and moreover that $$\iota_1^2={\rm id}$$ if $$\frac{r}{2}$$ is odd, but $$\iota_1^2=\iota_0$$ if $$\frac{r}{2}$$ is even. In either case, the subgroup $$\langle\iota_0,\iota_1\rangle$$ of $${\rm Aut}({\widetilde{\mathfrak S}_D(r)})$$ generated by $$\iota_0$$ and $$\iota_1$$ is commutative and contains four elements. We let $${\mathfrak S}_D(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_D(r)$$ by the action of $$\langle\iota_0,\iota_1\rangle$$. □ Definition. Let $$r$$ be even. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$D$$-symmetric if it admits a direct sum decomposition $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_{11}$$, $${\bf D}_{12}$$, $${\bf D}_{21}$$, $${\bf D}_{22}$$ satisfying the following conditions: (1) For all $$1\le i\le \frac{r}{2}-2$$ and all $$1\le s, t\le 2$$ we have $$k_i({\bf D}_{st})=k_{\frac{r}{2}+i}({\bf D}_{st})$$. (2) For all $$1\le i\le \frac{r}{2}-2$$ we have $$k_i({\bf D}_{11})=k_i({\bf D}_{12})$$ and $$k_i({\bf D}_{21})=k_i({\bf D}_{22})$$. (3) For $$j=1$$ and $$j=2$$ we have   k0(Dj1)=kr2(Dj2),kr2(Dj1)=k0(Dj2),kr2−1(Dj1)=kr−1(Dj2),kr−1(Dj1)=kr2−1(Dj2). (4)   ki(D11)=kr−i−1(D21) and ki(D12)=kr−i−1(D22) if $$i\in[0,r-1]$$ and $$\frac{r}{2}$$ is odd, or if $$i\in[1,\frac{r}{2}-2]\cup [\frac{r}{2}+1,r-2]$$ and $$\frac{r}{2}$$ is even. Moreover, if $$\frac{r}{2}$$ is even then   k0(D11)=kr−1(D21),kr2−1(D11)=k0(D21),kr2(D11)=kr2−1(D21),kr−1(D11)=kr2(D21). (5) $$\xi({\bf D}_{11})=\xi({\bf D}_{12})=\xi({\bf D}_{21})=\xi({\bf D}_{22})$$. (6) Modulo $$(p-1)$$ we have   s(Dj2)−s(Dj1)≡∑i=0r2−1ki(Dj1) for j=1,2s(D21)−s(D11)≡{r−24(kr2(D11)+k0(D11))+∑i=2r2−1(i−1)kr2−i(D11)r2 is odd(r4−1)kr2(D11)+r4k0(D11)+∑i=2r2−1(i−1)kr2−i(D11)r2 is even. (7) $$k_{\bullet}({\bf D}_{11})\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Lemma 2.6. Sending $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ to $$(n({\bf D}_{11}), s({\bf D}_{11}),\xi({\bf D}_{11}))$$ induces a bijection between the set of isomorphism classes of $$D$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_D(r)$$. □ Proof. Again we use Lemma 2.3. For a one-dimensional étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ put $$\alpha({\bf D})=(n({\bf D}), s({\bf D}),\xi({\bf D}))$$. If $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ is $$D$$-symmetric as above, then $$\alpha({\bf D}_{st})$$ is an element of $$\widetilde{\mathfrak S}_D(r)$$, for all $$1\le s,t\le2$$. Moreover, it is straightforward to check $$\iota_0(\alpha({\bf D}_{11}))=\alpha({\bf D}_{12})$$, $$\iota_0(\alpha({\bf D}_{21}))=\alpha({\bf D}_{22})$$, $$\iota_1(\alpha({\bf D}_{11}))=\alpha({\bf D}_{21})$$, and $$\iota_1(\alpha({\bf D}_{12}))=\alpha({\bf D}_{22})$$. It follows that the above map is well-defined and bijective.■ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$A$$-symmetric if $${\bf D}$$ admits a direct sum decomposition $${\bf D}=\oplus_{i=0}^{r-1}{\bf D}_i$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_i$$ satisfying   ki(Dj)=ki−j(D0),ξ(Dj)=ξ(D0),s(D0)−s(Dj)≡∑i=1jk−i(D0) modulo (p−1) for all $$i$$, $$j$$ (where we understand the sub index in $$k_{?}$$ as the unique representative in $$[0,r-1]$$ modulo $$r$$), and moreover   k∙(D0)∉{(0,…,0),(p−1,…,p−1)}. □ 3 Semi-infinite chamber galleries and the functor $${\bf D}$$ 3.1 Power multiplicative elements in the extended affine Weyl group Let $$G$$ be the group of $${\mathbb Q}_p$$-rational points of a $${\mathbb Q}_p$$-split connected reductive group over $${\mathbb Q}_p$$. Fix a maximal $${\mathbb Q}_p$$-split torus $$T$$ in $$G$$, let $$N(T)$$ be its normalizer in $$G$$. Let $$\Phi$$ denote the set of roots of $$T$$. For $$\alpha\in\Phi$$ let $$N_{\alpha}$$ be the corresponding root subgroup in $$G$$. Choose a positive system $$\Phi^+$$ in $$\Phi$$, let $$\Delta\subset\Phi^+$$ be the set of simple roots. Let $$N=\prod_{\alpha\in\Phi^+}N_{\alpha}$$. Let $$X$$ denote the semisimple Bruhat-Tits building of $$G$$, let $$A$$ denote its apartment corresponding to $$T$$. Our notational and terminological convention is that the facets of $$A$$ or $$X$$ are closed in $$X$$ (i.e., contain all their faces (the lower dimensional facets at their boundary)). A chamber is a facet of codimension $$0$$. For a chamber $$D$$ in $$A$$ let $$I_D$$ be the Iwahori subgroup in $$G$$ fixing $$D$$. We notice that $$I_D\cap N=\prod_{\alpha\in\Phi^+}I_{D}\cap N_{\alpha}$$. Fix a special vertex $$x_0$$ in $$A$$, let $$K$$ be the corresponding hyperspecial maximal compact open subgroup in $$G$$. Let $$T_0=T\cap K$$ and $$N_0=N\cap K$$. We have the isomorphism $$T/T_0\cong X_*(T)$$ sending $$\xi\in X_*(T)$$ to the class of $$\xi(p)\in T$$. Let $$I\subset K$$ be the Iwahori subgroup determined by $$\Phi^+$$. (If $${\rm red}:K\to\overline{K}$$ denotes the reduction map on to the reductive (over $${\mathbb F}_p)$$ quotient $$\overline{K}$$ of $$K$$, then $$I={\rm red}^{-1}({\rm red}(T_0N_0))$$.) Let $$C\subset A$$ be the chamber fixed by $$I$$. Thus $$I=I_C$$ and $$I\cap N=N_0=\prod_{\alpha\in\Phi^+}N_0\cap N_{\alpha}$$. We are interested in semi-infinite chamber galleries   C(0),C(1),C(2),C(3),… (3) in $$A$$ such that $$C=C^{(0)}$$ (and thus $$I=I_{C^{(0)}}$$) and such that, setting   N0(i)=IC(i)∩N, we have $$N_0=N_0^{(0)}$$ and   N0(0)⊃N0(1)⊃N0(2)⊃N0(3)⊃… with [N0(i):N0(i+1)]=p for all i≥0. (4) For any two chambers $$D_1\ne D_2$$ in $$A$$ sharing a common facet of codimension $$1$$ there is a uniquely determined $$\delta\in\Phi$$ with $$I_{D_1}\cap N_{\delta'}=I_{D_2}\cap N_{\delta'}$$ for $$\delta'\notin\{\delta,-\delta\}$$, with $$I_{D_1}\cap N_{-\delta}\subset I_{D_2}\cap N_{-\delta}$$ and with $$I_{D_2}\cap N_{\delta}\subset I_{D_1}\cap N_{\delta}$$, and both these inclusions are of index $$p$$. Moreover, the one-codimensional facet shared by $$D_1$$ and $$D_2$$ is contained in a translate of the hyperplane corresponding to $$\delta$$ (or equivalently, corresponding to $$-\delta$$).1 Applying this remark to $$D_1=C^{(i)}$$, $$D_2=C^{(i+1)}$$ and putting $$\alpha^{(i)}=\delta$$ we see that condition (4) precisely means that $$\alpha^{(i)}\in\Phi^+$$ for all $$i\ge0$$. Conversely, if condition (4) holds true then the sequence $$\alpha^{(0)},\alpha^{(1)},\alpha^{(2)},\alpha^{(3)},\ldots$$ in $$\Phi^+$$ can be characterized as follows: Setting   e[i,α]=|{0≤j≤i−1|α=α(j)}| for $$i\ge0$$ and $$\alpha\in\Phi^+$$, we have   N0(i)=∏α∈Φ+(N0∩Nα)pe[i,α]. Suppose that the center $$Z$$ of $$G$$ is connected. Then $$G/Z$$ is a semisimple group of adjoint type with maximal torus $$\check{T}=T/Z$$. Let $$\check{T}_0=T_0/(T_0\cap Z)\subset \check{T}$$. The extended affine Weyl group $$\widehat{W}=N(\check{T})/\check{T}_0$$ can be identified with the semidirect product between the finite Weyl group $$W=N(\check{T})/\check{T}=N(T)/T$$ and $$X_*(\check{T})$$. We identify $$A= X_*(\check{T})\otimes{\mathbb R}$$ such that $$x_0\in A$$ corresponds to the origin in the $${\mathbb R}$$-vector space $$X_*(\check{T})\otimes{\mathbb R}$$. We then regard $$\widehat{W}$$ as acting on $$A$$ through affine transformations. We regard $$\Delta\subset X^*({T})$$ as a subset of $$X^*(\check{T})$$. We usually enumerate the elements of $$\Delta$$ as $$\alpha_1,\ldots,\alpha_d$$, and we enumerate the corresponding simple reflection $$s_{\alpha}\in W$$ for $$\alpha\in\Delta$$ as $$s_1,\ldots,s_d$$ with $$s_i=s_{\alpha_i}$$. We assume that the root system $$\Phi$$ is irreducible. Let $${\alpha}_0\in\Phi$$ be the negative of the highest root. Let $$s_{\alpha_0}$$ be the corresponding reflection in the finite Weyl group $$W$$; define the affine reflection $$s_0=t_{\alpha_0^{\vee}}\circ s_{\alpha_0}\in \widehat{W}$$, where $$t_{\alpha_0^{\vee}}$$ denotes the translation by the coroot $$\alpha_0^{\vee}\in A$$ of $$\alpha_0$$. The affine Weyl group $$W_{{\rm aff}}$$ is the subgroup of $$\widehat{W}$$ generated by $$s_0,s_1,\ldots,s_d$$; in fact it is a Coxeter group with these Coxeter generators. The corresponding length function $$\ell$$ on $$W_{{\rm aff}}$$ extends to $$\widehat{W}$$. The above discussion shows that if the gallery (3) satisfies condition (4) and if $$w\in \widehat{W}$$ satisfies $$C^{(i)}=wC$$ for some $$i\ge0$$, then   pℓ(w)=pi=p[N0:IC(i)∩N0]. Let $$X_*(\check{T})_+$$ denote the set of dominant cocharacters. (Let $$T_+=\{t\in T\,|\,t N_0 t^{-1}\subset N_0\}$$, then $$X_*(\check{T})_+$$ is the image of $$T_+$$ under the map $$T_+\subset T\to T/T_0\cong X_*(T)\to X_*(\check{T})$$.) The monoid $$X_*(\check{T})_+$$ is free and has a distinguished basis $$\nabla$$, the set of fundamental coweights. The cone $${\mathcal C}$$ (vector chamber) in $$A$$ with origin in $$x_0$$ which is spanned by all the $$-\xi$$ for $$\xi\in \nabla$$ contains $$C$$, and $$C$$ is precisely the “top” chamber of this cone. The reflections $$s_0,s_1,\ldots,s_d$$ are precisely the reflections in the affine hyperplanes (walls) of $$A$$ which contain a codimension-$$1$$-face of $$C$$. Definition. We say that $$w\in \widehat{W}$$ is power multiplicative if $$\ell(w^m)=m\cdot\ell(w)$$ for all $$m\ge0$$. {(After finishing this article we found out that power multiplicative elements of $$\widehat{W}$$ have been studied, for example, in [3] under the name of straight elements.)} Of course, any element in the image of $$T\to N(\check{T})\to \widehat{W}=N(\check{T})/\check{T}_0$$ is power multiplicative because it acts on $$A$$ by translation (and $$\ell(w)$$ is the gallery distance between $$C$$ and $$wC$$). Suppose we are given a fundamental coweight $$\tau\in \nabla$$ and some non-trivial element $$\phi\in \widehat{W}$$ satisfying the following conditions: (a) $$\phi$$ is power multiplicative, (b) $$\tau$$ is minuscule, that is, we have $$\langle\alpha,\tau\rangle\in\{0,1\}$$ for all $$\alpha\in\Phi^+$$, (c) viewing $$\tau$$ via the embedding $$X_*(\check{T})\subset\widehat{W}$$ as an element of $$\widehat{W}$$, we have   ϕN∩τN≠∅. (5) □ Lemma 3.1. Let $$\phi$$ and $$\tau$$ be as above. Write $$\phi=\phi'v$$ with $$\phi'\in W_{{\rm aff}}$$ and $$v\in \widehat{W}$$ with $$vC=C$$. Choose a reduced expresssion   ϕ′=sβ(1)⋯sβ(r) of $$\phi'$$ with some function $$\beta:\{1,\ldots,r\}\to\{0,\ldots,d\}$$ (with $$r=\ell(\phi)=\ell(\phi')$$) and put   C(ar+b)=ϕasβ(1)⋯sβ(b)C (6) for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<r$$. Lift $$\tau\in \nabla\subset X_*(\check{T})$$ to some element of $$X_*({T})$$ and denote again by $$\tau$$ the corresponding homomorphism $${\mathbb Z}_p^{\times}\to T_0$$. Then we have: (i) The sequence   C=C(0),C(1),C(2),… satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (ii) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (iii) For any lifting $$\phi\in N(T)$$ of $$\phi\in \widehat{W}$$ we have $$\tau(a)\phi=\phi\tau(a)$$ in $$N(T)$$, for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. Given a chamber gallery (3) in $$A$$ with $$C=C^{(0)}$$ and $$N_0=N_0^{(0)}$$, condition (4) is equivalent with saying that (3) is a minimal chamber gallery in the cone $${\mathcal C}$$, that is, all $$C^{(i)}$$ are contained in $${\mathcal C}$$, and the gallery distance from $$C=C^{(0)}$$ to $$C^{(i)}$$ is $$i$$. Conversely, notice that if $$C^{(i)}$$ belongs to $${\mathcal C}$$ and the gallery distance from $$C=C^{(0)}$$ to $$C^{(i)}$$ is $$i$$, then also all $$C^{(j)}$$ for $$0\le j\le i$$ belong to $${\mathcal C}$$. (Otherwise the gallery would have to cross one of the bounding hyperplanes of $${\mathcal C}$$ and then cross it again later in the reverse direction; but a minimal gallery never a crosses a wall (hyperplane translate) back and forth.) Recall that, assuming condition (4), the set $$\{\alpha^{(j)}\,|\,j\ge0\}$$ is precisely the set of all $$\alpha\in\Phi^+$$ such that the gallery (3) crosses a translate of the hyperplane corresponding to $$\alpha$$. If there is some $$\epsilon\in X_*(T)$$ and some $$s\in{\mathbb N}$$ such that $$C^{(i+s)}=\epsilon C^{(i)}$$ for all $$i$$, then, since the action of $$\epsilon$$ is by translation on $$A$$, the latter set is the same as the set of all $$\alpha\in\Phi^+$$ such that the gallery $$C=C^{(0)},C^{(1)},\ldots, C^{(s)}$$ crosses a translate of the hyperplane corresponding to $$\alpha$$. Now let (3) be the chamber gallery defined by formula (6). (i) If $$n\in{\mathbb N}$$ is such that $$\phi^n\in X_*(T)$$, then the gallery distance between $$C$$ and $$\phi^nC$$ is $$\ell(\phi^n)$$. By construction, the gallery $$C=C^{(0)},C^{(1)},\ldots, C^{(n\ell(\phi))}=\phi^nC$$ has length $$n\ell(\phi)$$, that is, length $$\ell(\phi^n)$$ as $$\phi$$ is power multiplicative. Hence it must be a minimal chamber gallery. As $$\tau$$ is minuscule, it is in particular a dominant coweight. Therefore it follows from hypothesis (5) that also some power $$\phi^n$$ of $$\phi$$ is a dominant coweight. We thus obtain that $$C^{(n'\ell(\phi))}=\phi^{n'}C$$ lies in $${\mathcal C}$$ for each $$n'\in{\mathbb N}$$ divisible by $$n$$, but then, as the gallery is minimal and $${\mathcal C}$$ is a cone, $$C^{(i)}$$ lies in $${\mathcal C}$$ for each $$i\ge0$$. We get statement (i) in view of the preceeding discussion. (ii) The preceeding discussion shows $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,\langle\alpha,\phi^m\rangle\ne0\}$$ for any $$m\in{\mathbb N}$$ such that $$\phi^m$$ belongs to $$X_*(T)$$. As $$\phi$$ is power multiplicative, hypothesis (5) furthermore gives   {α(j)|j≥0}={α∈Φ+|⟨α,ϕm⟩≠0}={α∈Φ+|⟨α,τ⟩≠0} and as $$\tau$$ is minuscule this is the set   {α∈Φ+|⟨α,τ⟩=1}={α∈Φ+|α∘τ=idZp×}. (iii) By hypothesis (5) we have $$\tau^m=\phi^n$$ for some $$m,n\in{\mathbb N}$$. We deduce $$\tau^m=\phi\tau^m\phi^{-1}=(\phi\tau\phi^{-1})^{m}$$ and hence also $$\tau=\phi\tau\phi^{-1}$$ as $$\tau$$ and $$\phi\tau\phi^{-1}$$ belong to the free abelian group $$X_*(T)$$. Thus $$\tau\phi=\phi\tau$$ in $$\widehat{W}$$ which implies claim (iii). ■ Remark. For a given minuscule fundamental coweight $$\tau\in \nabla$$ any positive power $$\tau^m$$ of $$\tau$$ belongs to $$X_*(\check{T})$$, and so $$\phi=\tau^m$$ satisfies the assumptions of Lemma 3.1. However, for our purposes it is of interest to find $$\phi$$ (as in Lemma 3.1, possibly also required to project to $$W_{\rm aff}$$) of small length; the minimal positive power of $$\tau$$ belonging to $$X_*(\check{T})$$ is usually not optimal in this sense. □ 3.2 The functor D By $$I_0$$ we denote the pro-$$p$$-Iwahori subgroup contained in $$I$$. We often read $$\overline{T}=T_0/T_0\cap I_0$$ as a subgroup of $$T_0$$ by means of the Teichmüller character. Conversely, we read characters of $$\overline{T}$$ also as characters of $$T_0$$ (and do not introduce another name for these inflations). Let $${\rm ind}_{I_0}^{G}{\bf 1}_{{\mathfrak o}}$$ denote the $${\mathfrak o}$$-module of $${\mathfrak o}$$-valued compactly supported functions $$f$$ on $$G$$ such that $$f(ig)=f(g)$$ for all $$g\in G$$, all $$i\in I_0$$. It is a $$G$$-representation by means of $$(g'f)(g)=f(gg')$$ for $$g,g'\in G$$. Let   H(G,I0)=Endo[G](indI0G1o)op denote the corresponding pro-$$p$$-Iwahori Hecke algebra with coefficients in $${\mathfrak o}$$. For a subset $$H$$ of $$G$$ let $$\chi_H$$ denote the characteristic function of $$H$$. For $$g\in G$$ let $$T_g\in {\mathcal H}(G,I_0)$$ denote the Hecke operator corresponding to the double coset $$I_0gI_0$$. It sends $$f:G\to{\mathfrak o}$$ to   Tg(f):G⟶o,h↦∑x∈I0∖GχI0gI0(hx−1)f(x). Let $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ denote the category of $${\mathcal H}(G,I_0)$$-modules which as $${\mathfrak o}$$-modules are of finite length. We write $${\mathcal H}(G,I_0)_k={\mathcal H}(G,I_0)\otimes_{\mathfrak o}k$$. Given liftings $$\dot{s}\in N(T)$$ of all $$s\in S$$ we let $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ denote the $$k$$-subalgebra of $${\mathcal H}(G,I_0)_k$$ generated by the $$T_{\dot{s}}$$ for all $$s\in{S}$$ and the $$T_t$$ for $$t\in \overline{T}$$. Suppose we are given a reduced expression   ϕ=ϵs˙β(1)⋯s˙β(r) (7) (some function $$\beta:\{1,\ldots,r=\ell(\phi)\}\to\{0,\ldots,d\}$$, some $$\epsilon\in Z$$) of a power multiplicative element $$\phi\in N(T)$$, some power of which maps to a dominant coweight in $$N(T)/ZT_0$$. Put   C(ar+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<r$$. Then, by power multiplicativity of $$\phi$$, the sequence (3) thus defined satisfies property (4), cf. Lemma 3.1. Therefore we may use it to place ourselves into the setting (and notations) of [2], as follows. We define the half tree $$Y$$ whose edges are the elements in the $$N_0$$-orbits of the $$C^{(i)}\cap C^{(i+1)}$$ and whose vertices are the elements in the $$N_0$$-orbits of the $$C^{(i)}$$. We choose an isomorphism $$\Theta:Y\cong{\mathfrak X}_+$$ with the $$\lfloor{\mathfrak N}_0,\varphi,\Gamma\rfloor$$-equivariant half sub tree $${\mathfrak X}_+$$ of the Bruhat Tits tree of $${\rm GL}_2({\mathbb Q}_p)$$, satisfying the requirements of Theorem 3.2 of loc. cit. It sends the edge $$C^{(i)}\cap C^{(i+1)}$$ (respectively, the vertex $$C^{(i)}$$) of $$Y$$ to the edge $${\mathfrak e}_{i+1}$$ (respectively, the vertex $${\mathfrak v}_i$$) of $${\mathfrak X}_+$$. The half tree $$\overline{\mathfrak X}_+$$ is obtained from $${\mathfrak X}_+$$ by removing the “loose” edge $${\mathfrak e}_{0}$$. To an $${\mathcal H}(G,I_0)$$-module $$M$$ we associate the $$G$$-equivariant (partial) coefficient system $${\mathcal V}_M^X$$ on $$X$$. Briefly, its value at the chamber $$C$$ is $${\mathcal V}_M^X(C)=M$$. The transition maps $${\mathcal V}_M^X(D)\to{\mathcal V}_M^X(F)$$ for chambers (codimension-$$0$$-facets) $$D$$ and codimension-$$1$$-facets $$F$$ with $$F\subset D$$ are injective, and $${\mathcal V}_M^X(F)$$ for any such $$F$$ is the sum of the images of the $${\mathcal V}_M^X(D)\to{\mathcal V}_M^X(F)$$ for all $$D$$ with $$F\subset D$$. The pushforward $$\Theta_*{\mathcal V}_M$$ of the restriction of $${\mathcal V}_M^X$$ to $$Y$$ is a $$\lfloor{\mathfrak N}_0,\varphi^r,\Gamma_0\rfloor$$-equivariant coefficient system on $${\mathfrak X}_+$$ (and by further restriction on $$\overline{\mathfrak X}_+$$). This leads to the exact functor   M↦D(Θ∗VM)D(Θ∗VM)=H0(X¯+,Θ∗VM)⊗OE+OE=H0(X+,Θ∗VM)⊗OE+OE (8) from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{r},\Gamma_0)$$-modules over $${\mathcal O}_{\mathcal E}$$, where $$r=\ell(\phi)$$. If in addition we are given a homomorphism $$\tau:{\mathbb Z}_p^{\times}\to T_0$$ satisfying the conclusions of Lemma 3.1 (with respect to $$\phi$$), then this functor in fact takes values in the category of $$(\varphi^{r},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. For $$0\le i\le r-1$$ we put   yi=s˙β(1)⋯s˙β(i+1)s˙β(i)−1⋯s˙β(1)−1. Lemma 3.2. (a) $$y_i=y_{i-1}\cdots y_0\dot{s}_{\beta(i+1)}y_0^{-1}\cdots y_{i-1}^{-1}$$ and $$\phi=\epsilon y_{r-1}\cdots y_0$$. (b) $$y_i$$ is the affine reflection in the wall passing through $$C^{(i)}\cap C^{(i+1)}$$. □ Proof. To see (b) observe that $$y_i$$ indeed is a reflection, and that it sends $$C^{(i)}$$ to $$C^{(i+1)}$$. ■ 3.3 Supersingular modules For $$\alpha\in\Phi$$ we denote by $$\alpha^{\vee}$$ the associated coroot. For any $$\alpha\in\Phi$$ there is a corresponding homomorphism of algebraic groups $$\iota_{\alpha}:{\rm SL}_2({\mathbb Q}_p)\to G$$ as described in [5], Chapter II, Section 1.3. The element $$\iota_{\alpha}(\nu)$$ belongs to $$I\cap N_{\alpha}$$ and generates it as a topological group. For $$x\in{\mathbb F}_p^{\times}\subset {\mathbb Z}_p^{\times}$$ (via the Teichmüller character) we have $$\alpha^{\vee}(x)=\iota_{\alpha}(h(x))\in T$$. For $$0\le i\le d$$ let $$\overline{T}_{\alpha^{\vee}_i}$$ denote the subgroup of $$\overline{T}$$ given by the $${\mathbb F}_p$$-valued points of the schematic closure of $$\alpha^{\vee}_i(\overline{\mathbb F}_p^{\times})$$ (in the obvious $${\mathbb F}_p$$-group scheme (split torus) underlying $$\overline{T}$$). For a character $$\lambda:\overline{T}\to k^{\times}$$ let $$S_{\lambda}$$ be the subset of $$S=\{s_0,\ldots,s_d\}$$ consisting of all $$s_i$$ such that $$\lambda|_{\overline{T}_{\alpha^{\vee}_i}}$$ is trivial. Given $$\lambda$$ and a subset $${\mathcal J}$$ of $$S_{\lambda}$$ there is a uniquely determined character   χλ,J:H(G,I0)aff,k⟶k which sends $$T_t$$ to $$\lambda(t^{-1})$$ for $$t\in \overline{T}$$, which sends $$T_{\dot{s}}$$ to $$0$$ for $$s\in {S}-{\mathcal J}$$ and which sends $$T_{\dot{s}}$$ to $$-1$$ for $$s\in {\mathcal J}$$ (see [8] Proposition 2). For $$0\le i\le d$$ we define a number $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ such that   λ(αi∨(x))=xkiuad for allx∈Fp×, (9) as follows. If $$s_i\in{\mathcal J}$$ put $$k_i=p-1$$. Otherwise let $$k_i$$ be the unique integer in $$[0,p-2]$$ satisfying formula (9). Vignéras defined the notion of a supersingular$${\mathcal H}(G,I_0)_k$$-module, see [7, 8]. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-subalgebra of $${\mathcal H}(G,I_0)_k$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with all $$T_z$$ for $$z\in Z$$. Theorem 3.3. Vignéras, Ollivier) (a) Let $$\chi:{\mathcal H}(G,I_0)'_{{\rm aff},k}\to k$$ be a character extending $$\chi_{\lambda,{\mathcal J}}:{\mathcal H}(G,I_0)_{{\rm aff},k}\longrightarrow k$$ for some $$\lambda$$, $${\mathcal J}$$. If $$k_{\bullet}\notin\{(0,\ldots,0),(p-1,\ldots,p-1)\}$$ then the $${\mathcal H}(G,I_0)_k$$-module $${\mathcal H}(G,I_0)_k\otimes_{{\mathcal H}(G,I_0)'_{{\rm aff},k}}\chi$$ is supersingular. (b) Any supersingular $${\mathcal H}(G,I_0)_k$$-module of finite length contains a character $$\chi_{\lambda,{\mathcal J}}:{\mathcal H}(G,I_0)_{{\rm aff},k}\longrightarrow k$$ for some $$\lambda$$, $${\mathcal J}$$ with $$k_{\bullet}\notin\{(0,\ldots,0),(p-1,\ldots,p-1)\}$$. □ Proof. See [7] Theorem 5.14, Corollary 5.16, at least for the case where the root system underlying $$G$$ is irreducible. See [9] for more general statements. ■ It follows that the simple supersingular $${\mathcal H}(G,I_0)_k$$-modules are precisely the simple quotients of the $${\mathcal H}(G,I_0)_k$$-modules $${\mathcal H}(G,I_0)_k\otimes_{{\mathcal H}(G,I_0)'_{{\rm aff},k}}\chi$$ appearing in statement (a) of Theorem 3.3. We call the latter ones standard supersingular$${\mathcal H}(G,I_0)_k$$-modules. In fact, it is easy to see that these are simple themselves if the pair $$(k_{\bullet},{\mathcal J})$$ satisfies a certain non-periodicity property. 4 Classical matrix groups For $$m\in{\mathbb N}$$ let $$E_m\in{\rm GL}_m$$ denote the identity matrix and let $${E^*_d}$$ denote the standard antidiagonal element in $${\rm GL}_d$$ (i.e., the permutation matrix of maximal length). Let   S^m=(Em−Em),Sm=(EmEm). For $$1\le i,j\le m$$ with $$i\ne j$$ let $$\epsilon_{i,j}\in {\rm GL}_m$$ denote the matrix with entries $$1$$ on the diagonal and at the spot $$(i,j)$$ and with entries $$0$$ otherwise. 4.1 Affine root system $$\boldsymbol{\tilde{C}_d}$$} Assume $$d\ge2$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (s0s1)4=(sd−1sd)4=1 and (si−1si)3=1 for 2≤i≤d−1 (10) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and usiu=sd−i for 0≤i≤d. (11) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Consider the symplectic and the general symplectic group   Sp2d(Qp)={A∈GL2d(Qp)|TAS^dA=S^d},G=GSp2d(Qp)={A∈GL2d(Qp)|TAS^dA=κ(A)S^d for some κ(A)∈Qp×}. Let $$T$$ denote the maximal torus in $$G$$ consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T∩Sp2d(Qp)⟶Qp×,diag(x1,…,x2d)↦xi. For $$1\le i,j\le d$$ and $$\epsilon_1, \epsilon_2\in\{\pm 1\}$$ we thus obtain characters (using additive notation as usual) $$\epsilon_1e_i+\epsilon_2e_j:T\cap {\rm Sp}_{2d}({\mathbb Q}_p)\longrightarrow {\mathbb Q}_p^{\times}$$. We extend these latter ones to $$T$$ by setting   ϵ1ei+ϵ2ej:T⟶Qp×,A=diag(x1,…,x2d)↦xiϵ1xjϵ2κ(A)−ϵ1−ϵ22. For $$i=j$$ and $$\epsilon=\epsilon_1=\epsilon_2$$ we simply write $$\epsilon 2 e_i$$. Then $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}\cup\{\pm 2e_i\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$C_d$$. We choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}\cup\{2e_i\,|\,1\le i\le d\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=2e_d\}$$. The negative of the highest root is $$\alpha_0=-2e_1$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,Ed−1,x,Ed−1)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1)1≤i≤d−1diag(Ed−1,x,Ed−1,x−1)i=d. (12) For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm GL}_{2d}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S^1,Ed−i−1,Ei−1,S^1,Ed−i−1) and furthermore   s˙d=(Ed−11Ed−1−1),s˙0=(−p−1Ed−1pEd−1). Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $${\rm Sp}_{2d}({\mathbb Q}_p)\subset G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/ZT_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (19). Put   u˙=(Ed∗pEd∗). Then $$\dot{u}$$ belongs to $$N(T)$$ and normalizes $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/ZT_0$$ satisfies the relations (20). In $$N(T)$$ we consider   ϕ=(p⋅id)s˙ds˙d−1⋯s˙1s˙0. We may rewrite this as $$\phi=(p\cdot{\rm id})\dot{s}_{\beta(1)}\cdots\dot{s}_{\beta(d+1)}$$ where we put $$\beta(i)=d+1-i$$ for $$1\le i\le d+1$$. For $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<d+1$$ we put   C(a(d+1)+b)=ϕasd⋯sd−b+1C=ϕasβ(1)⋯sβ(b)C. Define the homomorphism   τ:Zp×⟶T0,x↦diag(xEd,Ed). Lemma 4.1. We have $$\phi^{d}\in T$$ and $$\phi^{d}N_0\phi^{-d}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For all $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) This is Lemma 3.1 in practice. A matrix computation shows $$\phi^{d}=(-1)^{d-1}{\rm diag}(p^{d+1}E_{d},p^{d-1}E_{d})\in T$$. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, by $$\epsilon_{i,i+d}$$ if $$\alpha=2e_i$$ with $$1\le i\le d$$, and by $$\epsilon_{i,j}\epsilon^{-1}_{i+d,j+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$. Using this we find   ϕdN0ϕ−d=∏α∈Φ+ϕd(N0∩Nα)ϕ−d=∏α∈Φ+(N0∩Nα)pmα,  mα={2α=ei+ej with 1≤i<j≤d2α=2ei with 1≤i≤d0all other α∈Φ+. In particular we find $$\phi^{d}N_0\phi^{-d}\subset N_0$$ and $$[N_0:\phi^{d}N_0\phi^{-d}]=p^{d(d+1)}$$. Let $$n\in {\mathbb N}$$. We have $$\ell(\phi^{n})\le{n(d+1)}$$ because the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$d+1$$ Coxeter generators. If $$m\ge n$$ is such that $$m\in d{\mathbb N}$$ we then have, on the other hand,   pℓ(ϕm)≥[I0:(I0∩ϕmIϕ−m)]≥[N0:(N0∩ϕmI0ϕ−m)]=[N0:ϕmN0ϕ−m]=pm(d+1). Thus   ℓ(ϕn)≥ℓ(ϕm)−ℓ(ϕm−n)≥m(d+1)−(m−n)(d+1)=n(d+1), hence $$\ell(\phi^{n})={n(d+1)}$$. We have shown that $$\phi$$ is power multiplicative and   [N0:ϕdN0ϕ−d]=[N0:(N0∩ϕdN0ϕ−d)]=[I0:(I0∩ϕdI0ϕ−d)]. We get $$I_0=N_0\cdot (I_0\cap\phi^{d}I_0\phi^{-d})$$ and that hypothesis (4) holds true. (b) As $$\phi^d\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation. ■ As explained in Section 3.2 we now obtain a functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{d+1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. As explained in [2], to compute it we need to understand the intermediate objects $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, acted on by $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma\rfloor$$. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{u}^{2}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ (see the proof of [8] Proposition 3). Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$2$$, a $$k$$-basis is $$e,f$$ where we write $$e=1\otimes e$$ and $$f=T_{\dot{u}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$0\le j\le d$$ put $$\widetilde{j}=d-j$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=(p⋅id)s˙β~(1)⋯s˙β~(d+1). Put $$n_e=\sum_{i=0}^{d}k_{d-i}p^i=\sum_{i=0}^{d}k_{\beta(i+1)}p^i$$ and $$n_f=\sum_{i=0}^{d}k_{i}p^i=\sum_{i=0}^{d}k_{\widetilde{\beta}(i+1)}p^i$$. Put $$\varrho=\prod_{i=0}^{d}(k_{i}!)=\prod_{i=0}^d(k_{\beta(i+1)}!)=\prod_{i=0}^d(k_{\widetilde{\beta}(i+1)}!)$$. Let $$0\le s_e, s_f\le p-2$$ be such that $$\lambda(\tau(x))=x^{-s_e}$$ and $$\lambda(\dot{u}\tau(x)\dot{u}^{-1})=x^{-s_f}$$ for all $$x\in{\mathbb F}_p^{\times}$$. Lemma 4.2. The assignment $$M[\lambda,{\mathcal J},b]\mapsto (n_e,s_e,b\varrho^{-1})$$ induces a bijection between the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_C(d+1)$$. □ Proof. We have $$\prod_{i=0}^d\alpha^{\vee}_{i}(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (12)). This implies   ∑i=0dki≡ne≡nf≡0 mod (p−1). (13) One can deduce from [8] Proposition 3 that for two sets of data $$\lambda, {\mathcal J}, b$$ and $$\lambda', {\mathcal J}', b'$$ the $${\mathcal H}(G,I_0)_k$$-modules $$M[\lambda,{\mathcal J},b]$$ and $$M[\lambda',{\mathcal J}',b']$$ are isomorphic if and only if $$b=b'$$ and the pair $$(\lambda, {\mathcal J})$$ is conjugate with the pair $$(\lambda', {\mathcal J}')$$ by means of a power of $$\dot{u}$$, that is, by means of $$\dot{u}^0=1$$ or $$\dot{u}^1=\dot{u}$$. Conjugating $$(\lambda,{\mathcal J})$$ by $$\dot{u}$$ has the effect of substituting $$k_{d-i}$$ with $$k_i$$, for any $$i$$. The datum of the character $$\lambda$$ is equivalent with the datum of $$s_e$$ together with all the $$k_i$$ taken modulo $$(p-1)$$ since the images of $$\tau$$ and all $$\alpha_i^{\vee}$$ together generate $$\overline{T}$$. Knowing the set $${\mathcal J}$$ is then equivalent with knowing the numbers $$k_i$$ themselves (not just modulo $$(p-1)$$). Thus, our mapping is well-defined and bijective. ■ Fix $$M=M[\lambda,{\mathcal J},b]$$. Let $$0\le j\le d$$. Normalize the homomorphism $$\iota_{\alpha_j}:{\rm SL}_2({\mathbb Q}_p)\to G$$ such that $$\iota_{\alpha_j}(\dot{s})=\dot{s}_j$$ for $$\dot{s}=\left( {0 \atop 1}\quad {{-1} \atop {0}}\right)$$. Let $$t_j=\iota_{\alpha_j}([\nu])-1\in k[[\iota_{\alpha_j}{\mathfrak N}_0]]\subset k[[N_0]]$$. Let $$F_j$$ denote the codimension-$$1$$-face of $$C$$ contained in the (affine) reflection hyperplane (in $$A\subset X$$) for $$s_j$$. Lemma 4.3. In $${\mathcal V}_M^X(F_j)$$ we have $$t_j^{k_j}\dot{s}_je=k_j!e$$ and $$t_j^{k_{d-j}}\dot{s}_jf=k_{d-j}!f$$ for all $$0\le j\le d$$. □ Proof. Let $$I_0^{j}$$ denote the subgroup of $$G$$ generated by $$I_0$$ and $$\dot{s}_{j}I_0 \dot{s}_{j}^{-1}$$. Let $$\overline{I}_0^{j}$$ denote the maximal reductive (over $${\mathbb F}_p$$) quotient of $$I_0^{j}$$. Then $$\iota_{\alpha_j}({\rm SL}_2({\mathbb Z}_p))\subset I_0^{j}$$ and $$\iota_{\alpha_j}$$ induces an isomorphism between either the quotient $$\overline{\mathcal S}={\rm SL}_2({\mathbb F}_p)$$ of $${\rm SL}_2({\mathbb Z}_p)$$, or the quotient $$\overline{\mathcal S}={\rm PSL}_2({\mathbb F}_p)$$ of $${\rm SL}_2({\mathbb Z}_p)$$, with the quotient $$\overline{I}_0^{j}$$ of $$I_0^{j}$$. The subgroup $$\overline{\mathcal U}$$ of $$\overline{\mathcal S}$$ generated by (the image of) $$\nu$$ maps isomorphically to the image of $$I_0$$ in $$\overline{I}_0^{j}$$. Therefore we get an embedding of $$k$$-algebras   H(S¯,U¯)k=Endk[S¯](indU¯S¯1k)op≅Endk[I0j](indI0I0j1k)op↪H(G,I0)aff,k⊂H(G,I0)k. Let $$\rho:{\mathcal H}(\overline{\mathcal S},\overline{\mathcal U})_k\to k$$ be the character defined by $$\rho(T_{\dot{s}})=-1$$ if $$k_{j}=p-1$$ but $$\rho(T_{\dot{s}})=0$$ if $$0\le k_{j}<p-1$$, and by $$\rho(T_{h(x)})=x^{-k_{j}}$$. We have a morphism of $${\rm SL}_2({\mathbb Z}_p)$$-representations (acting on $${\mathcal V}_M^X(F_j)$$ through $$\iota_{\alpha_j}$$)   (indU¯S¯1k)⊗H(S¯,U¯)kρ⟶VMX(Fj) sending a basis element $$e$$ of $$\rho$$ to $$e\in{\mathcal V}_M^X(F_j)$$. It is injective since its restriction to the space $$k.e$$ of $${\mathcal U}$$-invariants (see [2] Lemma 2.3) is injective. By [2] Lemma 2.5 we have $$t^{k_{j}}\dot{s}e=k_{j}!e$$ in $$({\rm ind}_{\overline{\mathcal U}}^{\overline{\mathcal S}}{\bf 1}_k)\otimes_{{\mathcal H}(\overline{\mathcal S},\overline{\mathcal U})_k}\rho$$ where $$t=[\nu]-1\in k[\overline{\mathcal U}]$$. We get $$t_j^{k_j}\dot{s}_je=k_j!e$$ in $${\mathcal V}_M^X(F_j)$$. A similar argument gives $$t_j^{k_{d-j}}\dot{s}_jf=k_{d-j}!f$$. For this notice that as the Hecke operator $$T_t$$ for $$t\in\overline{T}$$ acts on $$k.e$$ through $$\lambda(t^{-1})$$, it acts on $$k.f=k.T_{\dot{u}}e$$ through $$\lambda (\dot{u}t^{-1}\dot{u}^{-1})$$ (the same computation as in formula (18) below), and that formula (9) implies $$\lambda (\dot{u}\alpha_j^{\vee}(x)\dot{u}^{-1})=x^{k_{d-j}}$$. ■ Lemma 4.4. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tneφd+1e=ϱb−1e, (14)  tnfφd+1f=ϱb−1f, (15)  γ(x)e=x−see, (16)  γ(x)f=x−sff (17) for $$x\in{\mathbb F}_p^{\times}$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. The construction of $$\Theta$$ in [2] is based on the choice of elements $$\nu_i\in N_0$$ topologically generating the relevant pro-$$p$$ subgroup “at distance $$i$$’, for each $$0\le i\le d$$. These $$\nu_i$$ can be taken to be the elements $$y_{i-1}\cdots y_0\iota_{d-i}(\nu)y_0^{-1}\cdots y_{i-1}^{-1}$$. Their actions on the halftree $$Y$$ correspond to the actions of the $$\nu^{p^{i}}$$ on the halftree $${\mathfrak X}_+$$. We use the notations and the statements of Lemma 3.2, observing $$\beta(i+1)=d-i$$. For $$0\le i\le d$$ we have $$y_{i-1}\cdots y_0F_{d-i}=C^{(i)}\cap C^{(i+1)}$$ and $$y_{i-1}\cdots y_0C=C^{(i)}$$. Thus $$y_{i-1}\cdots y_0$$ defines isomorphisms   VMX(Fd−i)≅VMX(C(i)∩C(i+1)) and VMX(C)≅VMX(C(i)). But we have   VMX(C(i)∩C(i+1))=Θ∗VM(vi) and VMX(C(i))=Θ∗VM(ei). It follows that, under the above isomorphisms, the action of $$t_{d-i}$$, respectively, of $$\dot{s}_{d-i}$$, on $${\mathcal V}_M^X(F_{d-i})$$ becomes the action of $$[\nu]^{p^{i}}-1$$, respectively, of $$y_i$$, on $$\Theta_*{\mathcal V}_M({\mathfrak v}_i)$$. Now as we are in characteristic $$p$$ we have $$t^{p^i}=([\nu]-1)^{p^{i}}=[\nu]^{p^{i}}-1$$. Applying this to the element $$e$$, respectively, $$f$$, of $${\mathcal V}_M^X(C)\subset{\mathcal V}_M^X(F_{d-i})$$, Lemma 4.3 tells us   (tpi)kd−iyi⋯y0e=kd−i!yi−1⋯y0e, respectively, (tpi)kiyi⋯y0f=ki!yi−1⋯y0f. We compose these formulae for all $$0\le i\le d$$ and finally recall that the central element $$p\cdot{\rm id}$$ acts on $$M$$ through the Hecke operator $$T_{p^{-1}\cdot{\rm id}}$$, that is, by $$b^{-1}$$. We get formulae (15) and (14). Next recall that the action of $$\gamma(x)$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is given by that of $$\tau(x)\in T$$, that is, by the Hecke operator $$T_{\tau(x)^{-1}}$$. We thus compute   γ(x)e=Tτ(x)−1e=λ(τ(x))e,γ(x)f=Tτ(x)−1Tu˙e=Tu˙τ(x)−1e=Tu˙Tu˙τ(x)−1u˙−1e=Tu˙λ(u˙τ(x)u˙−1)e=λ(u˙τ(x)u˙−1)f (18) and obtain formulae (16) and (17).■ Corollary 4.5. The étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_e$$, $$g_f$$ such that   φd+1ge=bϱ−1tne+1−pd+1geφd+1gf=bϱ−1tnf+1−pd+1gfγ(x)ge−xsege∈t⋅kE+⋅geγ(x)gf−xsfgf∈t⋅kE+⋅gf. □ Proof. This follows from Lemma 4.4 as explained in [2] Lemma 6.4.■ Corollary 4.6. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. For $$x\in{\mathbb F}_p^{\times}$$ we have   τ(x)⋅u˙τ−1(x)u˙−1=diag(xEd,x−1Ed)=(∑i=0d(i+1)αi∨)(x) in $$\overline{T}$$. Applying $$\lambda$$ and observing $$\sum_{i=0}^dk_i\equiv 0$$ modulo $$(p-1)$$ we get   xsf−se=λ((∑i=0d(i+1)αi∨)(x))=x∑i=0diki and hence $$s_f-s_e\equiv \sum_{i=0}^dik_i$$ modulo $$(p-1)$$. Therefore   D(Θ∗VM)=D(Θ∗VM)1⊕D(Θ∗VM)2 with $${\bf D}(\Theta_*{\mathcal V}_M)_1=k_{\mathcal E}g_e$$ and $${\bf D}(\Theta_*{\mathcal V}_M)_2=k_{\mathcal E}g_f$$ is a direct sum decomposition of $${\bf D}(\Theta_*{\mathcal V}_M)$$ as an étale $$(\varphi^{d+1},\Gamma)$$-module over $$k_{\mathcal E}$$, and by Corollary (4.5) this decomposition identifies $${\bf D}(\Theta_*{\mathcal V}_M)$$ as being $$C$$-symmetric (recall in particular that $$n_e=\sum_{i=0}^{d}k_{d-i}p^i$$ and $$n_f=\sum_{i=0}^{d}k_{i}p^i$$). Composing this assignment $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ (from standard supersingular $${\mathcal H}(G,I_0)_k$$-modules to $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules) with the bijection of Lemma 2.5 (i) (from $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules to $${\mathfrak S}_C(d+1)$$) we get the bijection of Lemma 4.2 (from standard supersingular $${\mathcal H}(G,I_0)_k$$-modules to $${\mathfrak S}_C(d+1)$$). Thus, $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ is a bijection as stated.■ Remark. Consider the subgroup $$G'={\rm Sp}_{2d}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d},x^{-1}E_{d})$$ and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d}\dot{s}_{d-1}\cdots \dot{s}_{1}\dot{s}_0$$ then everything in fact happens inside $$G'$$. We then have $$\alpha^{(j)}\circ\tau={\rm id}^2_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which $$\tau(-1)$$ (i.e., $$T_{\tau(-1)}=T_{\tau(-1)^{-1}}$$) acts trivially. For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma^2\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^2=\{\gamma^2\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, following [2] (as a slight variation from what we explained in Section 3.2), we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{d+1},\Gamma^2)$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. In the case $$d=2$$ one may also work with $$\phi=(p\cdot{\rm id})\dot{s}_2\dot{s}_1\dot{s}_2\dot{u}$$. Its square is $$p\cdot{\rm id}$$ times the square of the $$\phi=(p\cdot{\rm id})\dot{s}_2\dot{s}_1\dot{s}_0$$ used above. □ 4.2 Affine root system $$\boldsymbol{\tilde{B}_d}$$ Assume $$d\ge 3$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (sd−1sd)4=1 and (s0s2)3=(si−1si)3=1 for 2≤i≤d−1 (19) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us0u=s1 and usiu=si for 2≤i≤d. (20) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Let   S~d=(Sd001)∈GL2d+1(Qp) and consider the special orthogonal group   G=SO2d+1(Qp)={A∈SL2d+1(Qp)|TAS~dA=S~d}. Let $$T$$ denote the maximal torus consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T⟶Qp×,diag(x1,…,xd,x1−1,…,xd−1,1)↦xi. Then (in additive notation) $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}\cup\{\pm e_i\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$B_d$$. We choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}\cup\{e_i\,|\,1\le i\le d\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=e_d\}$$. The negative of the highest root is $$\alpha_0=-e_1-e_2$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,x−1,Ed−2,x,x,Ed−2,1)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1,1)1≤i≤d−1diag(Ed−1,x2,Ed−1,x−2,1)i=d. (21) Remark. For roots $$\alpha\in\Phi$$ of the form $$\alpha=\pm e_i\pm e_j$$ the homomorphism $$\iota_{\alpha}:{\rm SL}_2\to {\rm SO}_{2d+1}$$ is injective. For roots $$\alpha\in\Phi$$ of the form $$\alpha=\pm e_i$$ the homomorphism $$\iota_{\alpha}:{\rm SL}_2\to {\rm SO}_{2d+1}$$ induces an embedding $${\rm PSL}_2\to {\rm SO}_{2d+1}$$. □ For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm SL}_{2d+1}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S1,Ed−i−1,Ei−1,S1,Ed−i−1,1) and furthermore   s˙d=(Ed−11Ed−11−1). Define   u˙=(p−1Ed−1pEd−1−1) and $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}$$. Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $$G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/T_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (19). Also $$\dot{u}$$ belongs to $$G$$; it normalizes $$T$$, $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/T_0=\widehat{W}$$ satisfies the relations (20). In $$N(T)$$ we consider the element   ϕ=s˙1s˙2⋯s˙d−1s˙ds˙d−1⋯s˙2s˙0. (22) We may rewrite this as $$\phi=\dot{s}_{\beta(1)}\cdots\dot{s}_{\beta(2d-1)}$$ where we put $$\beta(i)=i$$ for $$1\le i\le d$$ and $$\beta(i)=2d-i$$ for $$d\le i\le 2d-2$$ and $$\beta(2d-1)=0$$. We put   C(a(2d−1)+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<2d-1$$. Define the homomorphism   τ:Zp×⟶T0,x↦diag(x,Ed−1,x−1,Ed−1,1). Lemma 4.7. We have $$\phi^{2}\in T$$ and $$\phi^{2}N_0\phi^{-2}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) A matrix computation shows $$\phi^{2}={\rm diag}(p^2,E_{d-1},p^{-2},E_{d-1},1)$$. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon^{-1}_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, by $$\epsilon_{i,j}\epsilon^{-1}_{j+d,i+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$, and by $$\epsilon_{i,2d+1}\epsilon^{-1}_{2d+1,i+d}$$ if $$\alpha=e_i$$ with $$1\le i\le d$$. Using this we find   ϕ2N0ϕ−2=∏α∈Φ+ϕ2(N0∩Nα)ϕ−2=∏α∈Φ+(N0∩Nα)pmα,mα={2α=e1−ei with 1<i2α=e1+ei with 1<i2α=e10all other α∈Φ+. In particular we find $$\phi^{2}N_0\phi^{-2}\subset N_0$$ and $$[N_0:\phi^{2}N_0\phi^{-2}]=p^{2(2d-1)}$$. On the other hand, the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$2d-1$$ Coxeter generators. Arguing as in the proof of Lemma 4.1 we combine these facts to obtain $$\ell(\phi)=2d-1$$ and that $$\phi$$ is power multiplicative. We obtain that hypothesis (4) holds true, again by the same reasoning as in Lemma 4.1. (b) As $$\phi^{2}\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{2d-1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$ and a subset $${\mathcal J}\subset S_{\lambda}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. Notice that $$k_d$$ is necessarily even since in $$\alpha_d^{\vee}(x)$$ (formula (21)) the entry $$x$$ only appears squared. Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J]=H(G,I0)k⊗H(G,I0)aff,kk.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J}}$$. As a $$k$$-vector space, $$M$$ has dimension $$2$$, a $$k$$-basis is $$e,f$$ where we write $$e=1\otimes e$$ and $$f=T_{\dot{u}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J}]$$ for some $$\lambda, {\mathcal J}$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. A packet of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules is a set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules all of which give rise to the same $${\mathcal J}$$ and the same $$k_{\bullet}=(k_i)_{i}$$. □ Remark. Let $$\overline{T}'$$ denote the subgroup of $$\overline{T}$$ generated by the $$\alpha_i^{\vee}({\mathbb F}_p^{\times})$$ for all $$0\le i\le d$$. Formula (21) implies $$[\overline{T}:\overline{T}']=2$$. Two supersingular $${\mathcal H}(G,I_0)_k$$-modules $$M[\lambda,{\mathcal J}]$$ and $$M[\lambda',{\mathcal J}']$$ belong to the same packet if and only if $${\mathcal J}={\mathcal J}'$$ and if the restrictions of $$\lambda$$ and $$\lambda'$$ to $$\overline{T}'$$ coincide. □ For $$2\le j\le d$$ we put $$\widetilde{j}=j$$, furthermore we put $$\widetilde{0}=1$$ and $$\widetilde{1}=0$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=s˙β~(1)⋯s˙β~(2d−1). Put $$n_e=\sum_{i=0}^{2d-2}k_{\beta(i+1)}p^i$$ and $$n_f=\sum_{i=0}^{2d-2}k_{\widetilde{\beta}(i+1)}p^i$$. Put $$\varrho=k_0!k_1!k_d!\prod_{i=2}^{d-1}(k_{i}!)^2=\prod_{i=0}^{2d-2}(k_{\beta(i+1)}!)=\prod_{i=0}^{2d-2}(k_{\widetilde{\beta}(i+1)}!)$$. Let $$0\le s_e, s_f\le p-2$$ be such that $$\lambda(\tau(x))=x^{-s_e}$$ and $$\lambda(\dot{u}\tau(x)\dot{u}^{-1})=x^{-s_f}$$ for all $$x\in{\mathbb F}_p^{\times}$$. Lemma 4.8. The assignment $$M[\lambda,{\mathcal J}]\mapsto (n_e,s_e)$$ induces a bijection between the set of packets of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_B(2d-1)$$. □ Proof. We have $$\alpha^{\vee}_{0}(x)\alpha^{\vee}_{1}(x)\alpha^{\vee}_{{d}}(x)\prod_{i=2}^{d-1}(\alpha_{i}^{\vee})^2(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (21)). This implies   k0+k1+kd+2∑i=2d−1ki≡ne≡nf≡0 mod (p−1). (23) We further proceed exactly as in the proof of Lemma 4.2.■ Lemma 4.9. Let $$M=M[\lambda,{\mathcal J}]$$ for some $$\lambda$$, $${\mathcal J}$$. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tneφ2d−1e=ϱe, (24)  tnfφ2d−1f=ϱf, (25)  γ(x)e=x−see, (26)  γ(x)f=x−sff (27) for $$x\in{\mathbb F}_p^{\times}$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. As in Lemma 4.4. ■ Corollary 4.10. The étale $$(\varphi^{2d-1},\Gamma)$$-module over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_e$$, $$g_f$$ such that   φ2d−1ge=ϱ−1tne+1−p2d−1geφ2d−1gf=ϱ−1tnf+1−p2d−1gfγ(x)ge−xsege∈t⋅kE+⋅geγ(x)gf−xsfgf∈t⋅kE+⋅gf. □ Proof. This follows from Lemma 4.9 as explained in [2] Lemma 6.4.■ Corollary 4.11. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of packets of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$B$$-symmetric étale $$(\varphi^{2d-1},\Gamma)$$-modules $${\bf D}$$ over $$k_{\mathcal E}$$. □ Proof. For $$x\in{\mathbb F}_p^{\times}$$ we compute   τ(x)⋅u˙τ−1(x)u˙−1=diag(x2,Ed−1,x−2,Ed−1,1)=(α1∨−α0∨)(x) in $$\overline{T}$$. Application of $$\lambda$$ gives $$x^{s_f-s_e}=x^{k_1-k_0}$$ and hence $$s_f-s_e\equiv k_1-k_0=k_{\beta(1)}-k_{\beta(2d-1)}$$ modulo $$(p-1)$$. The required symmetry in the $$p$$-adic digits of $$n_e$$, $$n_f$$ is due to the corresponding symmetry of the function $$\beta$$. Thus, $${\bf D}(\Theta_*{\mathcal V}_M)$$ is a $$B$$-symmetric étale $$(\varphi^{2d-1},\Gamma)$$-module. One checks that composing $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ with the map of Lemma 2.5 (ii) gives the map of Lemma 4.8. As the maps in Lemmata 4.8 and 2.5 (ii) are bijective we obtain our result.■ 4.3 Affine root system $$\boldsymbol{\tilde{D}_d}$$ Assume $$d\ge4$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (sd−2sd)3=(s0s2)3=(si−1si)3=1 for 2≤i≤d−1 (28) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us0u=s1,us1u=s0,usd−1u=sd,usdu=sd−1,usiu=si for 2≤i≤d−2. (29) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with a four-element subgroup $$W_{\Omega}$$, with $$u\in W_{\Omega}$$ of order $$2$$.) Consider the general orthogonal group   GO2d(Qp)={A∈GL2d(Qp)|TASdA=κ(A)Sd for some κ(A)∈Qp×}. It contains the special orthogonal group   SO2d(Qp)={A∈SL2d(Qp)|TASdA=Sd}. Let $$G={\rm GSO}_{2d}({\mathbb Q}_p)$$ be the subgroup of $${\rm GO}_{2d}({\mathbb Q}_p)$$ generated by $${\rm SO}_{2d}({\mathbb Q}_p)$$ and by all $${\rm diag}(xE_d,E_d)$$ with $$x\in{\mathbb Q}_p^{\times}$$; it is of index $$2$$ in $${\rm GO}_{2d}({\mathbb Q}_p)$$. {(With the obvious definitions, $${\rm GO}_{2d}({\mathbb Q}_p)$$ is the group of $${\mathbb Q}_p$$-valued points of an algebraic group $${\rm GO}_{2d}$$, whereas $$G$$ is the group of $${\mathbb Q}_p$$-valued points of the connected component $${\rm GSO}_{2d}$$ of $${\rm GO}_{2d}$$; this $${\rm GSO}_{2d}$$ has connected center.) Let $$T$$ be the maximal torus consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T∩SL2d(Qp)⟶Qp×,diag(x1,…,x2d)↦xi. For $$1\le i,j\le d$$ and $$\epsilon_1, \epsilon_2\in\{\pm 1\}$$ we thus obtain characters (using additive notation as usual) $$\epsilon_1e_i+\epsilon_2e_j:T\cap {\rm SL}_{2d}({\mathbb Q}_p)\longrightarrow {\mathbb Q}_p^{\times}$$. We extend these latter ones to $$T$$ by setting   ϵ1ei+ϵ2ej:T⟶Qp×,A=diag(x1,…,x2d)↦xiϵ1xjϵ2κ(A)−ϵ1−ϵ22. Then $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$D_d$$. Choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=e_{d-1}+e_d\}$$. The negative of the highest root is $$\alpha_0=-e_1-e_2$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,x−1,Ed−2,x,x,Ed−2)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1)1≤i≤d−1diag(Ed−2,x,x,Ed−2,x−1,x−1)i=d. (30) For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm SL}_{2d}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S1,Ed−i−1,Ei−1,S1,Ed−i−1)=diag(Ei−1,S1,Ed−2,S1,Ed−i−1). Put   u˙=(p−1Ed−21pEd−21) and $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}$$ and $$\dot{s}_d=\dot{u}\dot{s}_{d-1}\dot{u}$$. Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $${\rm SO}_{2d}({\mathbb Q}_p)\subset G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/ZT_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (28). Also $$\dot{u}$$ belongs to $${\rm SO}_{2d}({\mathbb Q}_p)\subset G$$; it normalizes $$T$$, $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/ZT_0=\widehat{W}$$ satisfies the relations (29). In $$N(T)$$ we consider   ϕ=(p⋅id)s˙d−1s˙d−2⋯s˙2s˙1s˙ds˙d−2s˙d−3⋯s˙3s˙2s˙0 if d iseven,ϕ=(p2⋅id)s˙d−1s˙d−2⋯s˙2s˙1s˙ds˙d−2s˙d−3⋯s˙3s˙2s˙0 if d isodd. (The reason for our distinction according to the parity of $$d$$ is that the center $$Z$$ of $$G$$ is the subgroup of $$G$$ generated by $$T_0\cap Z$$ and $$p\cdot{\rm id}$$ if $$d$$ is even, respectively, by $$T_0\cap Z$$ and $$p^2\cdot{\rm id}$$ if $$d$$ is odd.) We may rewrite this as $$\phi=(p\cdot{\rm id})\dot{s}_{\beta(1)}\cdots \dot{s}_{\beta(2d-2)}$$ if $$d$$ is even, respectively, $$\phi=(p^2\cdot{\rm id})\dot{s}_{\beta(1)}\cdots \dot{s}_{\beta(2d-2)}$$ if $$d$$ is odd, where $$\beta(i)=d-i$$ for $$1\le i\le d-1$$, $$\beta(d)=d$$, $$\beta(i)=2d-1-i$$ for $$d+1\le i\le 2d-3$$ and $$\beta(2d-2)=0$$. We put   C(a(2d−2)+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<2d-2$$. Define the homomorphism   τ:Zp×⟶T0,x↦diag(xEd−1,Ed,x). Lemma 4.12. We have $$\phi^{d}\in T$$ and $$\phi^{d}N_0\phi^{-d}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) A matrix computation shows $$\phi^{d}={\rm diag}(p^{d+2}E_{d-1},p^{d-2}E_{d},p^{d+2})$$ if $$d$$ is even, and $$\phi^{d}={\rm diag}(p^{2d+2}E_{d-1},p^{2d-2}E_{d},p^{2d+2})$$ if $$d$$ is odd. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon^{-1}_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, and by $$\epsilon_{i,j}\epsilon^{-1}_{j+d,i+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$. Using this we find   ϕdN0ϕ−d=∏α∈Φ+ϕd(N0∩Nα)ϕ−d=∏α∈Φ+(N0∩Nα)pmα,mα={4α=ei+ej with 1≤i<j<d4α=ei−ed with 1≤i<d0all other α∈Φ+. In particular we find $$\phi^{d}N_0\phi^{-d}\subset N_0$$ and $$[N_0:\phi^{d}N_0\phi^{-d}]=p^{2d(d-1)}$$. On the other hand, the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$2d-2$$ Coxeter generators and of an element of length $$0$$. Arguing as in the proof of Lemma 4.1 we combine these facts to obtain $$\ell(\phi)=2d-2$$ and that $$\phi$$ is power multiplicative. We obtain that hypothesis (4) holds true, again by the same reasoning as in Lemma 4.1. (b) As $$\phi^{d}\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{2d-2},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Consider the elements   ω˙=(Ed∗pEd∗),ρ˙=(Ed−1∗ppEd−1∗1) of $${\rm GO}_{2d}({\mathbb Q}_p)$$. They normalize $$T$$ and satisfy   ω˙u˙=u˙ω˙,ω˙2=p⋅idω˙s˙iω˙−1=s˙d−i for 0≤i≤d,ρ˙2=p⋅u˙,ρ˙s˙iρ˙−1=s˙d−i for 2≤i≤d−2,ρ˙s˙d−1ρ˙−1=s˙1,ρ˙s˙dρ˙−1=s˙0,ρ˙s˙0ρ˙−1=s˙d−1,ρ˙s˙1ρ˙−1=s˙d. The element $$\dot{\omega}$$ belongs to $$G$$ if and only if $$d$$ is even. The element $$\dot{\rho}$$ belongs to $$G$$ if and only if $$d$$ is odd. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{\omega}^{2}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$ if $$d$$ is even, respectively, $$T_{p^2\cdot {\rm id}}=T_{\dot{\rho}^{4}}$$ and $$T^{-1}_{p^2\cdot {\rm id}}=T_{p^{-2}\cdot {\rm id}}$$ if $$d$$ is odd. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ if $$d$$ is even, respectively, which sends $$T_{p^2\cdot {\rm id}}$$ to $$b$$ if $$d$$ is odd (see the proof of [8] Proposition 3). We define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$4$$. A $$k$$-basis is $$e_0,e_1,f_0,f_1$$ where we write   e0=1⊗e,f0=Tu˙⊗e,e1=Tω˙⊗e,f1=Tu˙ω˙⊗e if d is even,e0=1⊗e,f0=Tu˙⊗e,e1=Tρ˙⊗e,f1=Tu˙ρ˙⊗e if d is odd. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$2\le j\le d-2$$ let $$\widetilde{j}=j$$, and furthermore let $$\widetilde{d-1}=d$$ and $$\widetilde{d}=d-1$$ and $$\widetilde{1}=0$$ and $$\widetilde{0}=1$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=(pn⋅id)s˙β~(1)⋯s˙β~(2d−2) with $$n=1$$ if $$d$$ is even, but $$n=2$$ if $$d$$ is odd. If $$d$$ is odd we consider in addition the following two maps $$\gamma$$ and $$\delta$$ from $$[1,2d-2]$$ to $$[0,d]$$. We put $$\gamma(1)=1$$, $$\gamma(d-1)=d$$, $$\gamma(d)=0$$, and $$\gamma(2d-2)=d-1$$. We put $$\delta(1)=0$$, $$\delta(d-1)=d-1$$, $$\delta(d)=1$$, and $$\delta(2d-2)=d$$. We put $$\gamma(i)=\delta(i)=\beta(2d-2-i)$$ for all $$i\in[2,\ldots,d-2]\cup[d+1,\ldots,2d-3]$$. We then have   ϱ˙ϕϱ˙−1=(p2⋅id)s˙γ(1)⋯s˙γ(2d−2),ϱ˙−1ϕϱ˙=(p2⋅id)s˙δ(1)⋯s˙δ(2d−2). Fix $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$. Put   ne0=∑i=02d−3kβ(i+1)pi,nf0=∑i=02d−3kβ~(i+1)pi forany parity of d,ne1=∑i=02d−3kβ(2d−2−i)pi,nf1=∑i=02d−3kβ~(2d−2−i)pi if d is even,ne1=∑i=02d−3kγ(i+1)pi,nf1=∑i=02d−3kδ(i+1)pi if d is odd. Let $$0\le s_{e_0}, s_{f_0}, s_{e_1}, s_{f_1}\le p-2$$ be such that for all $$x\in{\mathbb F}_p^{\times}$$ we have   λ(τ(x))=x−se0,λ(u˙τ(x)u˙−1)=x−sf0 forany parity of d,λ(ω˙τ(x)ω˙−1)=x−se1,λ(ω˙u˙τ(x)u˙−1ω˙−1)=x−sf1 if d is even,λ(ρ˙τ(x)ρ˙−1)=x−se1,λ(ρ˙u˙τ(x)u˙−1ρ˙−1)=x−sf1 if d is odd. Put $$\varrho=k_0!k_1!k_{d-1}!k_d!\prod_{i=2}^{d-2}(k_i!)^2=\prod_{i=0}^{2d-3}(k_{\beta(i+1)}!)=\prod_{i=0}^{2d-3}(k_{\widetilde{\beta}(i+1)}!)$$. Lemma 4.13. The assignment $$M[\lambda,{\mathcal J},b]\mapsto (n_{e_0}, s_{e_0}, b\varrho^{-1})$$ induces a bijection between the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_D(2d-2)$$. □ Proof. We have $$\alpha^{\vee}_{0}(x)\alpha^{\vee}_{1}(x)\alpha^{\vee}_{{d-1}}(x)\alpha^{\vee}_d(x)\prod_{i=2}^{d-2}(\alpha_{i}^{\vee})^2(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (30)). This implies   k0+k1+kd−1+kd+2∑i=2d−2ki≡ne0≡nf0≡ne1≡nf1≡0 mod (p−1). (31) It follows from [8] Proposition 3 that $$M[\lambda,{\mathcal J},b]$$ and $$M[\lambda',{\mathcal J}',b']$$ are isomorphic if and only if $$b=b'$$ and the pair $$(\lambda, {\mathcal J})$$ is conjugate with the pair $$(\lambda', {\mathcal J}')$$ by means of $$\dot{u}^{n}\dot{\omega}^m$$ for some $$n,m\in\{0,1\}$$ (if $$d$$ is even), respectively, by means of $$\dot{u}^{n}\dot{\rho}^m$$ for some $$n,m\in\{0,1\}$$ (if $$d$$ is odd). Under the map $$M[\lambda,{\mathcal J},b]\mapsto (n_{e_0}, s_{e_0}, b\varrho^{-1})$$, conjugation by $$\dot{u}$$ corresponds to the permutation $$\iota_0$$ of $$\widetilde{\mathfrak S}_D(2d-2)$$, while conjugation by $$\dot{\omega}$$, respectively, by $$\dot{\rho}$$, corresponds to the permutation $$\iota_1$$ of $$\widetilde{\mathfrak S}_D(2d-2)$$. We may thus proceed as in the proof of Lemma 4.2 to see that our mapping is well-defined and bijective.■ Lemma 4.14. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tnejφ2d−2ej=ϱb−1ej,tnfjφ2d−2fj=ϱb−1fj,γ(x)ej=x−sejej,γ(x)fj=x−sfjfj for $$x\in{\mathbb F}_p^{\times}$$ and $$j=0,1$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. As in Lemma 4.4.■ Corollary 4.15. The étale $$(\varphi^{2d-2},\Gamma)$$-module over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_{e_0}$$, $$g_{f_0}$$, $$g_{e_1}$$, $$g_{f_1}$$ such that for both $$j=0$$ and $$j=1$$ we have   φ2d−2gej=bϱ−1tnej+1−p2d−2gejφ2d−2gfj=bϱ−1tnfj+1−p2d−2gfjγ(x)(gej)−xsejgej∈t⋅kE+⋅gejγ(x)(gfj)−xsfjgfj∈t⋅kE+⋅gfj. □ Proof. This follows from Lemma 4.14 as explained in [2] Lemma 6.4.■ Corollary 4.16. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$D$$-symmetric étale $$(\varphi^{2d-2},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. We first verify that $${\bf D}(\Theta_*{\mathcal V}_M)$$ for $$M=M[\lambda,{\mathcal J},b]$$ is indeed $$D$$-symmetric. For this let $${\bf D}_{11}=\langle g_{e_0}\rangle$$, $${\bf D}_{12}=\langle g_{f_0}\rangle$$, $${\bf D}_{21}=\langle g_{e_1}\rangle$$, $${\bf D}_{22}=\langle g_{f_1}\rangle$$. Then $$k_i({\bf D}_{11})=k_{\beta(i+1)}$$ and $$k_i({\bf D}_{12})=k_{\widetilde{\beta}(i+1)}$$; moreover $$k_i({\bf D}_{21})=k_{\beta(2d-d-i)}$$ and $$k_i({\bf D}_{22})=k_{\widetilde{\beta}(2d-d-i)}$$ if $$d$$ is even, but $$k_i({\bf D}_{21})=k_{\gamma(i+1)}$$ and $$k_i({\bf D}_{22})=k_{\delta(i+1)}$$ if $$d$$ is odd. For the condition on $$s_{f_0}-s_{e_0}=s({\bf D}_{12})-s({\bf D}_{11})$$ we compute   τ(x)⋅u˙τ−1(x)u˙−1=diag(x,Ed−2,x−1,x−1,Ed−2,x)=(∑i=1d−1αi∨)(x), hence application of $$\lambda$$ gives $$x^{s_{f_0}-s_{e_0}}=x^{\sum_{i=1}^{d-1}k_i}$$ and hence $$s_{f_0}-s_{e_0}\equiv \sum_{i=1}^{d-1}k_i=\sum_{i=0}^{d-2}k_i({\bf D}_{11})$$ modulo $$(p-1)$$. The condition on $$s_{f_1}-s_{e_1}=s({\bf D}_{22})-s({\bf D}_{21})$$ in case $$d$$ is even is exactly verified like the one for $$s_{f_0}-s_{e_0}$$ because $$\dot{\omega}\tau(x)\dot{\omega}^{-1}\cdot\dot{\omega}\dot{u}\tau^{-1}(x)\dot{u}^{-1}\dot{\omega}^{-1}=\tau(x)\cdot\dot{u}\tau^{-1}(x)\dot{u}^{-1}$$. In case $$d$$ is odd the computation is   ρ˙τ(x)ρ˙−1⋅ρ˙u˙τ−1(x)u˙−1ρ˙−1=diag(x,Ed−2,x,x−1,Ed−2,x−1)=(αd∨+∑i=1d−2αi∨)(x), hence $$s_{f_1}-s_{e_1}\equiv k_d+\sum_{i=1}^{d-2}k_i=s({\bf D}_{12})-s({\bf D}_{11})+k_d-k_{d-1}=\sum_{i=0}^{d-2}k_i({\bf D}_{21})$$ modulo $$(p-1)$$. To see the condition on $$s_{e_1}-s_{e_0}=s({\bf D}_{21})-s({\bf D}_{11})$$ in case $$d$$ is even we compute   τ(x)⋅ω˙τ−1(x)ω˙−1=diag(1,xEd−2,1,1,x−1Ed−2,1)=(d−22αd−1∨+d−22αd∨+∑i=2d−2(i−1)αi∨)(x), (32) hence application of $$\lambda$$ gives $$x^{s_{e_1}-s_{e_0}}=x^{\frac{d-2}{2}k_{d-1}+\frac{d-2}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i}$$ and hence $$s_{e_1}-s_{e_0}\equiv \frac{d-2}{2}k_{d-1}+\frac{d-2}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i=\frac{d-2}{2}(k_{d-1}({\bf D}_{11})+k_{0}({\bf D}_{11}))+\sum_{i=2}^{d-2}(i-1)k_{d-i-1}({\bf D}_{11})$$ modulo $$(p-1)$$. If however $$d$$ is odd we compute   τ(x)⋅ρ˙τ−1(x)ρ˙−1=diag(1,xEd−2,x−1,1,x−1Ed−2,x)=(d−12αd−1∨+d−32αd∨+∑i=2d−2(i−1)αi∨)(x), hence application of $$\lambda$$ gives $$x^{s_{e_1}-s_{e_0}}=x^{\frac{d-1}{2}k_{d-1}+\frac{d-3}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i}$$ and hence $$s_{e_1}-s_{e_0}\equiv \frac{d-1}{2}k_{d-1}+\frac{d-3}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i=\frac{d-3}{2}k_{\frac{r}{2}}({\bf D}_{11})+\frac{d-1}{2}k_{0}({\bf D}_{11})+\sum_{i=2}^{d-2}(i-1)k_{d-i-1}({\bf D}_{11})$$ modulo $$(p-1)$$. We have shown that $${\bf D}(\Theta_*{\mathcal V}_M)$$ is $$D$$-symmetric. One checks that composing $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ with the map of Lemma 2.6 gives the map of Lemma 4.13. As the maps in Lemmata 4.13 and 2.6 are bijective we obtain our result.■ Remark. Consider the subgroup $$G'={\rm SO}_{2d}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d-1},x^{-1}E_{d},x)$$, and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d-1}\dot{s}_{d-2}\cdots \dot{s}_2\dot{s}_1\dot{s}_{d}\dot{s}_{d-2}\dot{s}_{d-3}\cdots \dot{s}_3\dot{s}_2\dot{s}_0$$, then everything in fact happens inside $$G'$$, and there is no dichotomy between $$d$$ even or odd. We then have $$\alpha^{(j)}\circ\tau={\rm id}^2_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which $$\tau(-1)$$ (i.e., $$T_{\tau(-1)}=T_{\tau(-1)^{-1}}$$) acts trivially. For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{2d-2},\Gamma^2\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^2=\{\gamma^2\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{2d-2},\Gamma^2)$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. Instead of the element $$\phi\in N(T)$$ used above we might also work with the element $$\dot{s}_{d-1}\cdots \dot{s}_2\dot{s}_1\dot{u}$$ of length $$d-1$$ (or products of this with elements of $$p^{\mathbb Z}\cdot{\rm id}$$), keeping the same $$C^{(\bullet)}$$. This results in a functor from $${\mathcal H}(G,I_0)$$-modules to $$(\varphi^{d-1},\Gamma)$$-modules. Up to a factor in $$p^{\mathbb Z}\cdot{\rm id}$$, the square of $$\dot{s}_{d-1}\cdots \dot{s}_2\dot{s}_1\dot{u}$$ is the element $$\phi$$ used above. □ Remark. For the affine root system of type $$D_d$$ there are three co minuscule fundamental coweights (cf. [1] Chapter 8, par 7.3]). The corresponding $$\phi$$’s for the other two choices are longer. □ 4.4 Affine root system $$\tilde{A}_d$$ Assume $$d\ge1$$ and consider $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$. Let   u˙=(Edp). For $$1\le i\le d$$ let   s˙i=diag(Ei−1,S1,Ed−i) and let $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}^{-1}$$. Let $$T$$ be the maximal torus consisting of diagonal matrices. Let $$\Phi^+$$ be such that $$N=\prod_{\alpha\in\Phi^+}N_{\alpha}$$ is the subgroup of upper triangular unipotent matrices. Let $$I_0$$ be the subgroup consisting of elements in $${\rm GL}_{d+1}({\mathbb Z}_p)$$ which are upper triangular modulo $$p$$. We put   ϕ=(p⋅id)s˙d⋯s˙0=(p⋅id)s˙β(1)⋯s˙β(d+1), where $$\beta(i)=d+1-i$$ for $$1\le i\le d+1$$. For $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<d+1$$ we put   C(a(d+1)+b)=ϕasd⋯sd−b+1C=ϕasβ(1)⋯sβ(b)C. We define the homomorphism   τ:Zp×⟶T0,x↦diag(Ed,x−1). The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). The corresponding $$\alpha^{(j)}\in\Phi^+$$ for $$j\ge0$$ satisfy $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$, and we have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. We thus obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{d+1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{u}^{d+1}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ (see the proof of [8] Proposition 3). Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$d+1$$, a $$k$$-basis is $$\{e_i\}_{0\le i\le d}$$ where we write $$e_i=T_{\dot{u}^{-i}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$0\le j\le d$$ put $$n_{e_j}=\sum_{i=0}^dk_{j-i}p^i$$ (reading $$j-i$$ as its representative modulo $$(d+1)$$ in $$[0,d]$$) and let $$s_{e_j}$$ be such that $$\lambda(\dot{u}^{-j}\tau(x)\dot{u}^j)=x^{-s_{e_j}}$$. Put $$\varrho=\lambda(-{\rm id})\prod_{i=0}^d(k_i!)$$. Theorem 4.17. The étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$\{g_{e_j}\}_{0\le j\le d}$$ such that for all $$j$$ we have   φd+1gej=bϱ−1tnej+1−pd+1gej,γ(x)gej−xsejgej∈t⋅kE+⋅gej. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$A$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. For the formulae describing $${\bf D}(\Theta_*{\mathcal V}_M)$$ one may proceed exactly as in the proof of Corollary 4.5. (The only tiny additional point to be observed is that the $$\dot{s}_i$$ (in keeping with our choice in [2]) do not lay in the images of the $$\iota_{\alpha_i}$$; this is accounted for by the sign factor $$\lambda(-{\rm id})$$ in the definition of $$\varrho$$.) Alternatively, as our $$\phi$$ is the $$(d+1)$$st power of the $$\phi$$ considered in Section 8 of [2], the computations of loc. cit. may be carried over. To see that $${\bf D}(\Theta_*{\mathcal V}_M)$$ is $$A$$-symmetric put $${\bf D}_j=\langle g_{e_j}\rangle$$ for $$0\le j\le d$$ and compare the above formulae with those defining $$A$$-symmetry; for example, we find $$s_{e_0}-s_{e_j}\equiv\sum_{i=1}^jk_i$$ modulo $$(p-1)$$. The bijectivity statement is then verified as before.■ Remark. Application of the functor of Lemma 2.1 to any one of the direct summands $${\bf D}_j$$ of the $$A$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ yields an étale $$(\varphi,\Gamma)$$-module isomorphic with the one assigned to $$M$$ in [2]. □ Remark. Consider the subgroup $$G'={\rm SL}_{d+1}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d},x^{-d})$$ and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d}\dot{s}_{d-1}\cdots \dot{s}_{1}\dot{s}_0$$ then everything in fact happens inside $$G'$$. We then have $$\alpha^{(j)}\circ\tau={\rm id}^{d+1}_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which the $$xE_{d+1}$$ (i.e., the $$T_{x^{-1}E_{d+1}}$$) for all $$x\in{\mathbb Z}_p^{\times}$$ with $$x^{d+1}=1$$ act trivially. (Notice that $$\tau(x)=xE_{d+1}$$ for such $$x$$.) For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma^{d+1}\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^{d+1}=\{\gamma^{d+1}\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{d+1},\Gamma^{d+1})$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. As all the fundamental coweights $$\tau$$ of $$T$$ are minuscule, each of them admits $$\phi$$’s for which the pair $$(\phi,\tau)$$ satisfies the properties asked for in Lemma 3.1. For example, let $$1\le g\le d$$. For $$\phi=\dot{s}_g\cdot \dot{s}_{g+1}\cdots \dot{s}_d\cdot \dot{u}$$ as well as for $$\phi=\dot{s}_g\cdot \dot{s}_{g-1}\cdots \dot{s}_1\cdot \dot{u}^{-1}$$ there is a unique minimal gallery from $$C$$ to $$\phi(C)$$ which admits a $$\phi$$-periodic continuation to a gallery (3), giving rise to a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{r},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$, where $$r=\ell(\phi)$$. □ 5 Exceptional groups of type $$\tilde{E}_6$$ and $$\tilde{E}_7$$ Let $$G$$ be the group of $${\mathbb Q}_p$$-rational points of a $${\mathbb Q}_p$$-split connected reductive group over $${\mathbb Q}_p$$ with connected center $$Z$$. Fix a maximal $${\mathbb Q}_p$$-split torus $$T$$ and define $$\Phi$$, $$N(T)$$, $$W$$, $$\widehat{W}$$, and $$W_{{\rm aff}}$$ as before. 5.1 Affine root system $$\tilde{E}_6$$ Assume that the root system $$\Phi$$ is of type $$E_6$$. Following [1] (for the indexing) we then have generators $$s_1,\ldots,s_6$$ of $$W$$ and $$s_0,\ldots,s_6$$ of $$W_{{\rm aff}}$$ such that   (s1s3)3=(s3s4)3=(s4s5)3=(s5s6)3=(s2s4)3=(s0s2)3=1 and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u3=1 and us4u−1=s4,us3u−1=s5,us5u−1=s2,us2u−1=s3,us1u−1=s6,us6u−1=s0,us0u−1=s1. (33) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the three-element subgroup $$W_{\Omega}=\{1,u,u^2\}$$.) Let $$e_1,\ldots,e_8$$ denote the standard basis of $${\mathbb R}^8$$. We use the standard inner product $$\langle.,.\rangle$$ on $${\mathbb R}^8$$ to view both the root system $$\Phi$$ as well as its dual $$\Phi^{\vee}$$ as living inside $${\mathbb R}^8$$. We choose a positive system $$\Phi^+$$ in $$\Phi$$ such that, as in [1], the simple roots are $$\alpha_1=\alpha_1^{\vee}=\frac{1}{2}(e_1+e_8-e_2-e_3-e_4-e_5-e_6-e_7)$$, $$\alpha_2=\alpha_2^{\vee}=e_2+e_1$$, $$\alpha_3=\alpha_3^{\vee}=e_2-e_1$$, $$\alpha_4=\alpha_4^{\vee}=e_3-e_2$$, $$\alpha_5=\alpha_5^{\vee}=e_4-e_3$$, $$\alpha_6=\alpha_6^{\vee}=e_5-e_4$$ while the negative of the highest root is $$\alpha_0=\alpha_0^{\vee}=\frac{1}{2}(e_6+e_7-e_1-e_2-e_3-e_4-e_5-e_8)$$. The set of positive roots is   Φ+={ej±ei|1≤i<j≤5}∪{12(−e6−e7+e8+∑i=15(−1)νiei)|∑i=15νi even}. We lift $$u$$ and the $$s_i$$ to elements $$\dot{u}$$ and $$\dot{s}_i$$ in $$N(T)$$. We then put   ϕ=s˙2s˙4s˙3s˙1u˙−1∈N(T). (34) We define $$\nabla$$ as in Section 3.1. Proposition 5.1. There is a $$\tau\in \nabla$$ such that the pair $$(\phi,\tau)$$ satisfies the hypotheses of Lemma 3.1. More precisely, $$\phi$$ is power multiplicative, and for the minuscule fundamental (co)weight $$\tau=\omega_1=\frac{2}{3}(e_8-e_7-e_6)\in \nabla$$ we have $$\phi^{12}=\tau^3$$ in $$W_{{\rm aff}}$$. □ Proof. (Here $$\tau^3$$ designates the three-fold iterate of translation by $$\tau=\omega_1$$, i.e., translation by the element $$3\omega_1$$ of the root lattice.) We define the subset   Φ(ω1)={12(−e6−e7+e8+∑i=15(−1)νiei)|∑i=15νi even} of $$\Phi^+$$. We compute $$\langle\beta,3\omega_1\rangle=3$$ for each $$\beta\in \Phi(\omega_1)$$, but $$\langle\beta,3\omega_1\rangle=0$$ for each $$\beta\in \Phi^+-\Phi(\omega_1)$$. This means that for each $$\beta\in \Phi(\omega_1)$$ the translation by $$3\omega_1$$ crosses $$3$$ walls parallel to the hyperplane corresponding to $$\beta$$, and that for each $$\beta\in \Phi^+-\Phi(\omega_1)$$ it crosses no wall parallel to the hyperplane corresponding to $$\beta$$. As $$\Phi(\omega_1)$$ contains 16 elements it follows that $$\ell(\tau^3)=\ell(3\omega_1)=3\cdot 16=48$$. On the other hand, in the Appendix we explain how one can prove $$\phi^{12}=3\omega_1$$ (as elements in $$W_{{\rm aff}}$$). Combining these two facts we deduce $$\ell(\phi^{12})=48$$ and hence that $$\phi$$ (which a priori has length $$\le 4$$) is power multiplicative.■ Remark. Instead of verifying $$\phi^{12}=3\omega_1$$ by means of a direct (computer) computation, one might try to prove that the affine transformation $$\phi^{12}$$ on $$A$$ crosses exactly the same walls as does $$3\omega_1$$ (which are described in the above proof). □ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{4},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Similarly, we may replace $$\phi$$ by its third power $$\phi^{3}$$ which (in contrast to $$\phi$$) is an element of $$W_{\rm aff}$$ (modulo $$T_0$$). It yields a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{12},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. As in our treatment of the cases $$C$$, $$B$$, $$D$$, and $$A$$, this functor identifies the set of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules bijectively with a set of certain $$E$$-symmetric étale $$(\varphi^{12},\Gamma)$$-modules over $${k}_{\mathcal E}$$ of dimension $$3$$. We leave the details to the reader. Remarks. (a) Dual to the above choice of $$\phi$$ is the choice   ϕ=s˙2s˙4s˙5s˙6u˙∈N(T). (35) For this choice, Proposition 5.1 holds true verbatim the same way, but now with the minuscule fundamental (co)weight $$\tau=\omega_6=\frac{1}{3}(3e_5+e_8-e_7-e_6)$$ with its corresponding subset (again containing $$16$$ elements)   Φ(ω6)={12(e5−e6−e7+e8+∑i=14(−1)νiei)|∑i=14νi even}∪{e5±ei|1≤i<5} of $$\Phi^+$$. Again see the Appendix. (b) For $$\phi$$ given by either (34) or (35), consider the corresponding reduced expression of $$\phi^3$$ in $$W_{{\rm aff}}$$ (obtained by three-fold concatenation and then conjugation of the $$s_i$$’s appearing with powers of $$u$$ so that no factor $$u$$ or $$u^2$$ remains). The number of occurrences of the $$s_i$$ are precisely the coefficients of the $$\alpha^{\vee}_i$$ in   α0∨+α1∨+α6∨+2α2∨+2α3∨+2α5∨+3α4∨=0. □ 5.2 Affine root system $$\tilde{E}_7$$ Assume that the root system $$\Phi$$ is of type $$E_7$$. Following [1] we then have generators $$s_1,\ldots,s_7$$ of $$W$$ and $$s_0,\ldots,s_7$$ of $$W_{{\rm aff}}$$ such that   (s0s1)3=(s1s3)3=(s3s4)3=(s4s5)3=(s5s6)3=(s6s7)3=(s2s4)3=1 and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us4u=s4,us2u=s2,us3u=s5,us6u=s1,us7u=s0,us5u=s3,us1u=s6,us0u=s7. (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Let $$e_1,\ldots,e_8$$ denote the standard basis of $${\mathbb R}^8$$. We use the standard inner product $$\langle.,.\rangle$$ on $${\mathbb R}^8$$ to view both the root system $$\Phi$$ as well as its dual $$\Phi^{\vee}$$ as living inside $${\mathbb R}^8$$. We choose a positive system $$\Phi^+$$ in $$\Phi$$ such that, as in [1], the simple roots are $$\alpha_1=\alpha_1^{\vee}=\frac{1}{2}(e_1+e_8-e_2-e_3-e_4-e_5-e_6-e_7)$$, $$\alpha_2=\alpha_2^{\vee}=e_2+e_1$$, $$\alpha_3=\alpha_3^{\vee}=e_2-e_1$$, $$\alpha_4=\alpha_4^{\vee}=e_3-e_2$$, $$\alpha_5=\alpha_5^{\vee}=e_4-e_3$$, $$\alpha_6=\alpha_6^{\vee}=e_5-e_4$$, $$\alpha_7=\alpha_7^{\vee}=e_6-e_5$$ while the negative of the highest root is $$\alpha_0=\alpha_0^{\vee}=e_7-e_8$$. The set of positive roots is   Φ+={ej±ei|1≤i<j≤6}∪{e8−e7}∪{12(e8−e7+∑i=16(−1)νiei)|∑i=16νi odd}. We lift $$u$$ and the $$s_i$$ to elements $$\dot{u}$$ and $$\dot{s}_i$$ in $$N(T)$$. We then put   ϕ=s˙1s˙3s˙4s˙2s˙5s˙4s˙3s˙1s˙0u˙∈N(T). We define $$\nabla$$ as in Section 3.1. Proposition 5.2. There is a $$\tau\in \nabla$$ such that the pair $$(\phi,\tau)$$ satisfies the hypotheses of Lemma 3.1. More precisely, $$\phi$$ is power multiplicative, and for the minuscule fundamental (co)weight $$\tau=\omega_7=e_6+\frac{1}{2}(e_8-e_7)\in \nabla$$ we have $$\phi^{6}=\tau^2$$ in $$W_{{\rm aff}}$$. □ Proof. Exactly the same as for Proposition 5.1. The corresponding subset in $$\Phi^+$$ is   {12(e6+e8−e7+∑i=15(−1)νiei)|∑i=15νi odd}∪{e8−e7}∪{e6±ei|1≤i<6}. It contains 27 elements, thus $$\ell(2\omega_7)=54$$. For a computer proof of $$\phi^{6}=2\omega_7$$ see the Appendix.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{9},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Similarly, we may replace $$\phi$$ by its square $$\phi^{2}$$ which (in contrast to $$\phi$$) is an element of $$W_{\rm aff}$$ (modulo $$T_0$$). It yields a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{18},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Again this functor identifies the set of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules bijectively with a set of certain $$E$$-symmetric étale $$(\varphi^{18},\Gamma)$$-modules over $${k}_{\mathcal E}$$ of dimension $$2$$. We leave the details to the reader. Remark. The number of occurrences of the $$s_i$$ in $$\phi^2\in W_{{\rm aff}}$$ (cf. the case $$\tilde{E}_6$$) are the coefficients of the $$\alpha^{\vee}_i$$ in   α0∨+α7∨+2(α1∨+α2∨+α6∨)+3(α3∨+α5∨)+4α4∨=0. □ Appendix A.1 Verification of the statement $$\phi^{12}=3\omega_1$$ in the proof of Proposition 5.1 In the computer algebra system sage, the input R$$\,{=}\,$$RootSystem([“E”,6,1]).weight_lattice() Lambda$$\,{=}\,$$R.fundamental_weights() omega1$$\,{=}\,$$Lambda[1]-Lambda[0] R.reduced_word_of_translation(3*omega1) prompts the output   [0,2,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1]. (36) By definition of the function reduced_word_of_translation this means $$s^*_{i_1}\cdots s^*_{i_{48}}=3\omega_1$$, with the string $$[i_1,\ldots,i_{48}]$$ as given by (A.1). Here $$s_i^*=s_i$$ for $$1\le i\le 6$$, but $$s_0^*$$ denotes the affine reflection in the outer face of Bourbaki’s fundamental alcove $$A$$. Since we deviate from these conventions in that our $$s_0$$ is the affine reflection in the outer face of the negative$$C=-A$$ of $$A$$, we must modify the above string (A.1) as follows. First, writing $$s_i^{**}=s_0^*s_i^*s_0^*$$ for $$0\le i\le 6$$, conjugating the factors in the previous word by $$s_0^*$$ and commuting some of its factors where allowed, the above says $$s^{**}_{j_1}\cdots s^{**}_{j_{48}}=3\omega_1$$ where the string $$[j_1,\ldots,j_{48}]$$ is given by   [2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1]. (37) The $$s_i^{**}$$ are precisely the reflections in the codimension $$1$$ faces of $$s_0^*A$$. But $$s_0^*A$$ is a translate of $$C$$, and under this translation, the reflection $$s_0^{**}=s_0^*$$ corresponds to $$s_0$$, whereas for $$1\le i\le 6$$ the reflection $$s_i^{**}$$ corresponds to $$w_0s_iw_0$$, where $$w_0$$ is the longest element of $$W$$. We have $$w_0s_iw_0=s_i$$ for $$i\in\{0,2,4\}$$, but $$w_0s_3w_0=s_5$$ and $$w_0s_1w_0=s_6$$. Thus, we obtain $$s_{k_1}\cdots s_{k_{48}}=3\omega_1$$ where the string $$[k_1,\ldots,k_{48}]$$ is obtained from the string (A.2) by keeping its entry values $$0, 2$$, and $$4$$, while exchanging the entry values $$3$$ with $$5$$ and $$1$$ with $$6$$. Using formulae (33) one checks that $$s_{k_1}\cdots s_{k_{48}}=\phi^{12}$$. $$\Box$$ A.2 Verification of the statement $$\boldsymbol{\phi^{12}=3\omega_6}$$ for $$\boldsymbol{\phi}$$ given by (35) The argument is the same as in Proposition 5.1. The string returned by sage to R$$\,{=}\,$$RootSystem([“E”,6,1]).weight_lattice(), Lambda$$\,{=}\,$$R.fundamental_weights(), omega6$$\,{=}\,$$Lambda[6]-Lambda[0], R.reduced_word_of_translation(3*omega6) reads   [0,2,4,3,1,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,3,4,5,6]. A.3 Verification of the statement $$\boldsymbol{\phi^{6}=2\omega_7}$$ in the proof of Proposition 5.2 The argument is the same as in Proposition 5.1. The string returned by sage to R$$\,{=}\,$$RootSystem([“E”,7,1]).weight_lattice(), Lambda$$\,{=}\,$$R.fundamental_weights(), omega7$$\,{=}\,$$Lambda[7]-Lambda[0], R.reduced_word_of_translation(2*omega7) reads   [0,1,3,4,2,5,4,3,1,0,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,0,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,5,4,2,3,4,5,6,7]. (38) Acknowledgments I would like to thank Laurent Berger for a helpful discussion related to this work. I am very grateful to the referees for the careful reading of the manuscript and the detailed suggestions for improvements. Footnotes 1 (Pick $$g\in N(T)$$ acting as the reflection in the affine hyperplane which contains the one-codimensional facet shared by $$D_1$$ and $$D_2$$. Then $$g D_1=D_2$$ and hence $$I_{D_2}=g I_{D_1}g^{-1}$$, so we can read off all these claims.) References [1] Bourbaki N. Groupes at Algèbres de Lie, ch. 4, 5, 6 / ch. 7, 8, 9 . Springer-Verlag, Berlin Heidelberg, 2006. [2] Grosse-Klönne E. “From pro-$$p$$-Iwahori Hecke modules to $$(\Phi,\Gamma)$$-modules, I.” Duke Mathematical Journal  165, no. 8 ( 2016): 1529– 95. [3] He X. and Nie. S. “Minimal length elements of extended affine Weyl groups.” Compositio Mathematica  150, no. 11 ( 2014): 1903– 27. Google Scholar CrossRef Search ADS   [4] Iwahori N. and Matsumoto. H. “On some Bruhat decomposition and the structure of the Hecke rings of $$p$$-adic Chevalley groups.” Publications Mathèmatiques de lĺHèS  25 ( 1965): 5– 48. Google Scholar CrossRef Search ADS   [5] Jantzen J. C. Representations of Algebraic Groups . Mathematical Surveys and Monographs 107, 2nd ed. American Mathematical Society, USA, 2003. [6] Koziol K. “Pro-$$p$$-Iwahori invariants for $${\rm SL}_2$$ and $$L$$-packets of Hecke modules.” International Mathematics Research Notices  2016, no. 4 ( 2016): 1090- 125. Google Scholar CrossRef Search ADS   [7] Ollivier R. “Compatibility between Satake and Bernstein-type isomorphisms in characteristic $$p$$.” Algebra and Number Theory  5 ( 2014): 1071– 111. Google Scholar CrossRef Search ADS   [8] Vignèras M. F. “Pro-$$p$$-Iwahori Hecke ring and supersingular $$\overline{\mathbb F}_p$$-representations.” Mathematische Annalen  331, no. 3 ( 2005): 523- -56 and Mathematische Annalen 333, no. 3 ( 2005): 699– 701. Google Scholar CrossRef Search ADS   [9] Vignèras M. F. “The pro-$$p$$-Iwahori Hecke algebra of a reductive $$p$$-adic group III.” Journal of the Institute of Mathematics of Jussieu  (to appear). © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Mathematics Research Notices Oxford University Press

From pro- $$p$$ Iwahori–Hecke Modules to $$(\varphi,\Gamma)$$ -Modules, II

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Abstract

Abstract Let $${\mathfrak o}$$ be the ring of integers in a finite extension field of $${\mathbb Q}_p$$, let $$k$$ be its residue field. Let $$G$$ be a split reductive group over $${\mathbb Q}_p$$, let $${\mathcal H}(G,I_0)$$ be its pro-$$p$$ Iwahori–Hecke $${\mathfrak o}$$-algebra. In [2] we introduced a general principle how to assign to a certain additionally chosen datum $$(C^{(\bullet)},\phi,\tau)$$ an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from finite length $${\mathcal H}(G,I_0)$$-modules to $$(\varphi^r,\Gamma)$$-modules. In this paper we concretely work out such data $$(C^{(\bullet)},\phi,\tau)$$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular $${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$$-modules with the set of $$(\varphi^r,\Gamma)$$-modules satisfying a certain symmetry condition. 1 Introduction Let $${\mathfrak o}$$ be the ring of integers in a finite extension field of $${\mathbb Q}_p$$, let $$k$$ be its residue field. Let $$G$$ be a split reductive group over $${\mathbb Q}_p$$, let $$T$$ be a maximal split torus in $$G$$, let $$I_0$$ be a pro-$$p$$-Iwahori subgroup fixing a chamber $$C$$ in the $$T$$-stable apartment of the semisimple Bruhat Tits building of $$G$$. Let $${\mathcal H}(G,I_0)$$ be the pro-$$p$$-Iwahori Hecke $${\mathfrak o}$$-algebra. Let $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ denote the category of $${\mathcal H}(G,I_0)$$-modules of finite $${\mathfrak o}$$-length. From a certain additional datum $$(C^{(\bullet)},\phi,\tau)$$ we constructed in [2] an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of étale $$(\varphi^r,\Gamma)$$-modules (with $$r\in{\mathbb N}$$ depending on $$\phi$$). For $$G={\rm GL}_2({\mathbb Q}_p)$$, when precomposed with the functor of taking $$I_0$$-invariants, this yields the functor from smooth $${\mathfrak o}$$-torsion representations of $${\rm GL}_2({\mathbb Q}_p)$$ (or at least from those generated by their $$I_0$$-invariants) to étale $$(\varphi,\Gamma)$$-modules which plays a crucial role in Colmez’ construction of a $$p$$-adic local Langlands correspondence for $${\rm GL}_2({\mathbb Q}_p)$$. In [2] we studied in detail the functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ when $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ for $$d\ge1$$. In [6] the situation has been analyzed for $$G={\rm SL}_{d+1}({\mathbb Q}_p)$$. The purpose of this paper is to explain how the general construction of [2] can be installed concretely for other classical matrix groups $$G$$ (as well as for $$G$$’s of type $$E_6$$, $$E_7$$). Recall that $$C^{(\bullet)}=(C=C^{(0)}, C^{(1)},C^{(2)},\ldots)$$ is a minimal gallery, starting at $$C$$, in the $$T$$-stable apartment, that $$\phi\in N(T)$$ is a “period” of $$C^{(\bullet)}$$ and that $$\tau$$ is a homomorphism from $${\mathbb Z}_p^{\times}$$ to $$T$$, compatible with $$\phi$$ in a suitable sense. The above $$r\in{\mathbb N}$$ is just the length of $$\phi$$. It turns out that $$\tau$$ must be a minuscule fundamental coweight (at least if the underlying root system is simple). Conversely, any minuscule fundamental coweight $$\tau$$ can be included into a datum $$(C^{(\bullet)},\phi,\tau)$$, in such a way that some power of $$\tau$$ is a power of $$\phi$$. For $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ we gave explicit choices of $$(C^{(\bullet)},\phi,\tau)$$ with $$r=1$$ in [2] (there are essentially just two choices, and these are “dual” to each other). In this paper we work out “privileged” choices $$(C^{(\bullet)},\phi,\tau)$$ for the classical matrix groups, as well as for $$G$$’s of type $$E_6$$, $$E_7$$. We mostly consider $$G$$ with connected center $$Z$$. Our choices of $$(C^{(\bullet)},\phi,\tau)$$ are such that $$\phi\in N(T)$$ projects (modulo $$ZT_0$$, where $$T_0$$ denotes the maximal bounded subgroup of $$T$$) to the affine Weyl group (viewed as a subgroup of $$N(T)/ZT_0$$). In particular, up to modifications by elements of $$Z$$ these $$\phi$$ can also be included into data $$(C^{(\bullet)},\phi,\tau)$$ for the other $$G$$’s with the same underlying root system, not necessarily with connected center. We indicate these modifications along the way. Notice that the $$\phi\in N(T)$$ considered in [2] for $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$ does not project to the affine Weyl group, only its $$(d+1)$$st power (which is considered here) does so. But since the discussion is essentially the same, our treatment of the case $$A$$ here is very brief. In either case we work out the behavior of the functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ on those $${\mathcal H}(G,I_0)_k={\mathcal H}(G,I_0)\otimes_{\mathfrak o}k$$-modules which we call “standard supersingular”. Roughly speaking, these are induced from characters of the pro-$$p$$-Iwahori Hecke algebra of the corresponding simply connected group. Each irreducible supersingular $${\mathcal H}(G,I_0)_k$$-module is contained in (and in “most” cases is equal to) a standard supersingular $${\mathcal H}(G,I_0)_k$$-module (and the very few standard supersingular $${\mathcal H}(G,I_0)_k$$-modules which are not irreducible supersingular are easily identified). We show that our functor induces a bijection between the set of isomorphism classes (respectively, certain packets of such if $$G={\rm SO}_{2d+1}({\mathbb Q}_p)$$) of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and the set of (isomorphism classes of) étale $$(\varphi^r,\Gamma)$$-modules over $$k_{\mathcal E}=k((t))$$ which satisfy a certain symmetry condition (depending on the root system underlying $$G$$). They are direct sums of one-dimensional étale $$(\varphi^r,\Gamma)$$-modules, their dimension is the $$k$$-dimension of the corresponding standard supersingular $${\mathcal H}(G,I_0)_k$$-module. The interest in étale $$(\varphi^r,\Gamma)$$-modules lies in their relation with $${\rm Gal}_{{\mathbb Q}_p}$$-representations. For any $$r\in{\mathbb N}$$ there is an exact functor from the category of étale $$(\varphi^r,\Gamma)$$-modules to the category of étale $$(\varphi,\Gamma)$$-modules (it multiplies the rank by the factor $$r$$), and by means of Fontaine’s functor, the latter one is equivalent with the category of $${\rm Gal}_{{\mathbb Q}_p}$$-representations. In [2] we also explained that a datum $$(C^{(\bullet)},\phi)$$ alone, that is, without a $$\tau$$ as above, can be used to define an exact functor $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of étale $$(\varphi^r,\Gamma_0)$$-modules, where $$\Gamma_0$$ denotes the maximal pro-$$p$$-subgroup of $$\Gamma\cong{\mathbb Z}_p^{\times}$$. Such data $$(C^{(\bullet)},\phi)$$ are not tied to co minuscule coweights and exist in abundance. We do not discuss them here. We hope that, beyond its immediate purposes as described above, this paper may also be a useful reference for explicit descriptions of the pro-$$p$$-Iwahori algebra (in particular with respect to the various Weyl groups involved) of classical matrix groups other than $${\rm GL}_d$$ (we could not find such descriptions in the literature). The outline is as follows. In Section 2 we explain the functor from étale $$(\varphi^r,\Gamma)$$-modules to étale $$(\varphi,\Gamma)$$-modules, and we introduce the “symmetric” étale $$(\varphi^r,\Gamma)$$-modules mentioned above, for each of the root systems $$C$$, $$B$$, $$D$$, and $$A$$. In Section 3, Lemma 3.1, we discuss the relation between the data $$(C^{(\bullet)},\phi,\tau)$$ and minuscule fundamental coweights. Our discussions of classical matrix groups $$G$$ in Section 4 are just concrete incarnations of Lemma 3.1, although in neither of these cases there is a need to make formal reference to Lemma 3.1. On the other hand, in our discussion of the cases $$E_6$$ and $$E_7$$ in Section 5 we do invoke Lemma 3.1. We tried to synchronize our discussions of the various matrix groups. As a consequence, arguments repeat themselves, and we do not write them out again and again. In the Appendix we record calculations relevant for the cases $$E_6$$ and $$E_7$$, carried out with the help of the computer algebra system sage. 2 $$\boldsymbol{(\varphi^r,\Gamma)}$$-modules We often regard elements of $${\mathbb F}_p^{\times}$$ as elements of $${\mathbb Z}_p^{\times}$$ by means of the Teichmüller lifting. In $${\rm GL}_2({\mathbb Z}_p)$$ we define the subgroups   Γ=(Zp×001),Γ0=(1+pZp001),N0=(1Zp01) and the elements   φ=(p001),ν=(1101),h(x)=(x00x−1),γ(x)=(x001), where $$x\in {\mathbb Z}_p^{\times}$$. Let $${\mathcal O}_{\mathcal E}^+={\mathfrak o}[[{\mathfrak N}_0]]$$ denote the completed group ring of $${\mathfrak N}_0$$ over $${\mathfrak o}$$. Let $${\mathcal O}_{\mathcal E}$$ denote the $$p$$-adic completion of the localization of $${{{\mathcal O}_{\mathcal E}^+}}$$ with respect to the complement of $$\pi_K{{{\mathcal O}_{\mathcal E}^+}}$$, where $$\pi_K\in{\mathfrak o}$$ is a uniformizer. In the completed group ring $$k_{\mathcal E}^+=k[[{\mathfrak N}_0]]$$ we put $$t=[\nu]-1$$. Let $$k_{\mathcal E}={\rm Frac}(k_{\mathcal E}^+)={{{\mathcal O}_{\mathcal E}}}\otimes_{{\mathfrak o}}k$$. We identify $$k_{\mathcal E}^+=k[[t]]$$ and $$k_{\mathcal E}=k((t))$$. For definitions and notational conventions concerning étale $$\varphi^r$$-modules and étale $$(\varphi^r,\Gamma)$$-modules we refer to [2]. Let $$r\in{\mathbb N}$$. Let $${\bf D}=({\bf D},{\varphi}^r_{{\bf D}})$$ be an étale $$\varphi^r$$-module over $${\mathcal O}_{\mathcal E}$$. For $$0\le i\le r-1$$ let $${\bf D}^{(i)}={\bf D}$$ be a copy of $${\bf D}$$. For $$1\le i\le r-1$$ define $$\varphi_{\widetilde{\bf D}}:{\bf D}^{(i)}\to{\bf D}^{(i-1)}$$ to be the identity map on $${\bf D}$$, and define $$\varphi_{\widetilde{\bf D}}:{\bf D}^{(0)}\to{\bf D}^{(r-1)}$$ to be the structure map $${\varphi}^r_{{\bf D}}$$ on $${\bf D}$$. Together we obtain a $${\mathbb Z}_p$$-linear endomorphism $$\varphi_{\widetilde{\bf D}}$$ on   D~=⨁i=0r−1D(i). Define an $${\mathcal O}_{\mathcal E}$$-action on $$\widetilde{\bf D}$$ by the formula $$x\cdot((d_i)_{0\le i\le r-1})=(\varphi^i_{{\mathcal O}_{\mathcal E}}(x)d_i)_{0\le i\le r-1}$$. Then the endomorphism $$\varphi_{\widetilde{\bf D}}$$ of $$\widetilde{\bf D}$$ is semilinear with respect to this $${\mathcal O}_{\mathcal E}$$-action, hence it defines on $$\widetilde{\bf D}$$ the structure of an étale $$\varphi$$-module over $${\mathcal O}_{\mathcal E}$$, see [2], Section 6.3. Let $$\Gamma'$$ be an open subgroup of $$\Gamma$$, let $${\bf D}$$ be an étale $$(\varphi^r,\Gamma')$$-module over $${\mathcal O}_{\mathcal E}$$. Define an action of $$\Gamma'$$ on $$\widetilde{\bf D}$$ by   γ⋅((di)0≤i≤r−1)=(γ⋅di)0≤i≤r−1. Lemma 2.1. The $$\Gamma'$$-action on $$\widetilde{\bf D}$$ commutes with $$\varphi_{\widetilde{\bf D}}$$ and is semilinear with respect to the $${\mathcal O}_{\mathcal E}$$-action, hence we obtain on $$\widetilde{\bf D}$$ the structure of an étale $$(\varphi,\Gamma')$$-module over $${\mathcal O}_{\mathcal E}$$. We thus obtain an exact functor from the category of étale $$(\varphi^r,\Gamma')$$-modules to the category of étale $$(\varphi,\Gamma')$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Proof. This is immediate from the respective properties of the $$\Gamma'$$-action on $${\bf D}$$. ■ The conjugation action of $$\Gamma$$ on $${\mathfrak N}_0$$ allows us to define for each $$x\in {\mathbb Z}_p^{\times}$$ an automorphism $$f\mapsto \gamma(x)f\gamma(x^{-1})$$ of $$k[[t]]=k_{\mathcal E}^+$$. Lemma 2.2. For $$x\in {\mathbb F}_p^{\times}$$ and $$m, n\in{\mathbb Z}_{\ge0}$$ we have   γ(x)tnprmγ(x−1)−(xt)nprm∈t(n+1)prmk[[t]]. (1) □ Proof. Denote again by $$x$$ be the representative of $$x$$ in $$[1,p-1]$$. We compute $$[\nu]^x-1=\sum_{j=1}^x{x\choose j}([\nu]-1)^x=\sum_{j=1}^x{x\choose j}t^j$$, hence   γ(x)tγ(x−1)−xt=([ν]x−1)−xt∈t2k[[t]],γ(x)tnγ(x−1)−(xt)n∈tn+1k[[t]],◼γ(x)tnprmγ(x−1)−(xt)nprm=(γ(x)tnγ(x−1)−(xt)n)prm∈t(n+1)prmk[[t]]. ■ Lemma 2.3. (a) Let $${\bf D}$$ be a one-dimensional étale $$(\varphi^r,\Gamma)$$-module over $$k_{{\mathcal E}}$$. There exists a basis element $$g$$ for $${\bf D}$$, uniquely determined integers $$0\le s({\bf D})\le p-2$$ and $$1\le n({\bf D})\le p^r-1$$ and a uniquely determined scalar $$\xi({\bf D})\in k^{\times}$$ such that   φDrg=ξ(D)tn(D)+1−prgγ(x)g−xs(D)g∈t⋅kE+⋅g for all $$x\in{\mathbb Z}_p^{\times}$$. Thus, one may define $$0\le k_{i}({\bf D})\le p-1$$ by $$n({\bf D})=\sum_{i=0}^{r-1}k_{i}({\bf D})p^i$$. One has $$n({\bf D})\equiv 0$$ modulo $$(p-1)$$. (b) For any given integers $$0\le s\le p-2$$ and $$1\le n\le p^r-1$$ with $$n\equiv 0$$ modulo $$(p-1)$$ and any scalar $$\xi\in k^{\times}$$ there is a uniquely determined (up to isomorphism) one-dimensional étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ with $$s=s({\bf D})$$ and $$n=n({\bf D})$$ and $$\xi=\xi({\bf D})$$. □ Proof. (a) Begin with an arbitrary basis element $$g_0$$ for $${\bf D}$$; then $$\varphi_{{\bf D}}^rg_0=F g_0$$ for some unit $$F\in k((t))=k_{{\mathcal E}}$$. After multiplying $$g_0$$ with a suitable power of $$t$$ we may assume $$F=\xi t^{m}(1+t^{n_0}F_0)$$ for some $$0\ge m\ge 2-p^r$$, some $$\xi\in k^{\times}$$, some $$n_0>0$$, and some $$F_0\in k[[t]]$$ (use $$t^{p^r}\varphi^r=\varphi^rt$$). For $$g_1=(1+t^{n_0}F_0)g_0$$ we then get $$\varphi_{{\bf D}}^rg_1=\xi t^{m}(1+t^{n_1}F_1)g_1$$ for some $$n_1>n_0>0$$ and some $$F_1\in k[[t]]$$. We put $$g_2=(1+t^{n_1}F_1)g_1$$. We may continue in this way. In the limit we get, by completeness, $$g=g_{\infty}\in{\bf D}$$ such that $$\varphi_{{\bf D}}^rg=\xi t^{m}g$$. It is clear that $$\xi({\bf D})=\xi$$ and $$n({\bf D})=p^{r}-1+m$$ are well-defined. Next, as $$\gamma(x)^{p-1}$$ (for $$x\in{\mathbb Z}_p^{\times}$$) is topologically nilpotent (in the topological group $$\Gamma$$) and acts by an automorphism on $${\bf D}$$, there is some unit $$F_x\in k[[t]]$$ with $$\gamma(x)g=F_xg$$. Thus, the action of $$\Gamma$$ on $${\bf D}$$ induces an action of $$\Gamma$$ on $$k[[t]]/tk[[t]]\cong k$$, and this action is given by a homomorphism $${\mathbb Z}_p^{\times}\cong\Gamma\to k^{\times}$$, $$x\mapsto x^{s({\bf D})}$$ for some $$0\le s({\bf D})\le p-2$$. Computing modulo $$t\cdot k^+_{{\mathcal E}}\cdot g$$ we have (with $$n=n({\bf D})$$, $$s=s({\bf D})$$, $$\xi=\xi({\bf D})$$)   xsg≡γ(x)g=ξ−1γ(x)tpr−n−1φDrg=ξ−1(γ(x)tpr−n−1γ(x−1))γ(x)φDrg≡(i)ξ−1(xt)pr−n−1γ(x)φDrg≡xsξ−1(xt)pr−n−1φDrg=xs+pr−n−1g. Here $$(i)$$ follows from $$\gamma(x)t^{p^r-n-1}\gamma(x^{-1})-(xt)^{p^r-n-1}\in t^{p^r-n}k[[t]]$$, formula (1). In short, we have $$g\equiv x^{p^r-n-1}g$$ for all $$x\in{\mathbb F}_p^{\times}$$. We deduce $$n\equiv0$$ modulo $$p-1$$. (b) Put $$D=k[[t]]=k^+_{\mathcal E}$$ and $$D^*={\rm Hom}_k^{\rm ct}(k^+_{\mathcal E},k)$$. Endow $$D^*$$ with an action of $$k^+_{\mathcal E}$$ by putting $$(\alpha\cdot\ell)(x)=\ell(\alpha\cdot x)$$ for $$\alpha\in k^+_{\mathcal E}$$, $$\ell\in D^*$$ and $$x\in D$$. For $$j\ge0$$ define $$\ell_j\in D^*$$ by   ℓj(∑i≥0aiti)=aj. Then $$\{\ell_j\}_{j\ge0}$$ is a $$k$$-basis of $$D^*$$, and we have $$t\ell_{j+1}=\ell_j$$ and $$t\ell_0=0$$. Define $$\varphi^r:D^*\to D^*$$ to be the $$k$$-linear map with $$\varphi^r(\ell_j)=\xi^{-1}\ell_{p^rj+n}$$. We then find   (tpr(φr(ℓj)))(∑i≥0aiti)=(ξ−1ℓprj+n)(∑i≥0aiti+pr)=ξ−1apr(j−1)+n,(φr(tℓj))(∑i≥0aiti)=ξ−1ℓpr(j−1)+n(∑i≥0aiti)=ξ−1apr(j−1)+n, (with $$a_i=0$$, respectively, $$\ell_i=0$$ for $$i<0$$). As $$t^{p^r}\varphi^r=\varphi^rt$$ in $$k^+_{\mathcal E}[\varphi^r]$$ this means that we have defined an action of $$k^+_{\mathcal E}[\varphi^r]$$ on $$D^*$$. Next, for $$x\in {\mathbb F}_p^{\times}$$ and $$b,m\in{\mathbb Z}_{\ge0}$$ we put   γ(x)⋅ℓ−b+n∑i=0m−1pri=x−s(γ(x)tbγ(x−1))ℓ−b+n∑i=0m−1pri, or equivalently,   γ(x)⋅tb(φr)mℓ0=x−s(γ(x)tbγ(x−1))(φr)mℓ0. (2) We claim that this defines an action of $$\gamma$$ on $$D^*$$. In order to check well definedness, that is, independence on the choice of $$b$$ and $$m$$ when only $$-b+n\sum_{i=0}^{m-1}p^{ri}$$ is given, it is enough to consider the equations $$t^b(\varphi^r)^m\ell_0=\xi t^{b+np^{rm}}(\varphi^r)^{m+1}\ell_0$$ and to compare the respective effect of $$\gamma(x)$$ (through formula (2)) on either side. Namely, we need to see   ξ−1x−s(γ(x)tbγ(x−1))(φr)mℓ0=x−s(γ(x)tb+nprmγ(x−1))(φr)m+1ℓ0, or equivalently,   (γ(x)tbγ(x−1))(ξ−1−(γ(x)tnprmγ(x−1))φr)(φr)mℓ0=0. For this it is enough to see that $$\xi^{-1}-(\gamma(x)t^{np^{rm}}\gamma(x^{-1})\varphi^r)$$ annihilates $$(\varphi^r)^m\ell_0$$. Now $$n<p^r$$ implies $$(n+1)p^{rm}> n\sum_{i=0}^mp^{ri}$$ and hence that $$t^{(n+1)p^{rm}}$$ annihilates $$(\varphi^r)^{m+1}\ell_0=\xi^{-m-1}\ell_{n\sum_{i=0}^mp^{ri}}$$. Therefore it is enough, by formula (1), to show that $$\xi^{-1}-(xt)^{np^{rm}}\varphi^r$$ annihilates $$(\varphi^r)^m\ell_0$$. But $$n\equiv 0$$ modulo $$(p-1)$$ means $$x^{np^{rm}}=1$$. Or claim follows since $$\xi^{-1}(\varphi^r)^m\ell_0=t^{np^{rm}}(\varphi^r)^{m+1}\ell_0$$. In fact, we have defined an action of $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$ on $$D^*$$. Namely, it is clear from the definitions that the relations $$\gamma(x)\cdot t= (\gamma(x)t\gamma(x^{-1}))\cdot \gamma(x)$$ in $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$ are respected. That the actions of $$\gamma(x)$$ and $$\varphi^r$$ on $$D^*$$ commute follows easily from the relations $$\gamma(x)\varphi^r=\varphi^r\gamma(x)$$ and $$t^{p^r}\varphi^r=\varphi^rt$$ in $$k^+_{\mathcal E}[\varphi^r,\Gamma]$$. Notice that   γ(x)ℓ0=x−sℓ0 for all x∈Zp×. Now passing to the dual $$D\cong (D^*)^*$$ of $$D^*$$ yields a non-degenerate $$(\psi^r,\Gamma)$$-module over $$k^+_{\mathcal E}$$ with an associated étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ with $$s=s({\bf D})$$ and $$n=n({\bf D})$$ and $$\xi=\xi({\bf D})$$; this is explained in [2] Lemma 6.4. This dualization argument also proves the uniqueness of $${\bf D}$$. ■ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$C$$-symmetric if it admits a direct sum decomposition $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_1$$, $${\bf D}_2$$ satisfying the following conditions (1), (2C), and (3C): (1) $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$0\le i\le r-1$$. (2C) $$\xi({\bf D}_1)=\xi({\bf D}_2)$$. (3C) $$s({\bf D}_2)-s({\bf D}_1)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_1)$$ modulo $$(p-1)$$. (4) $$k_{\bullet}({\bf D}_1)\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$B$$-symmetric if $$r$$ is odd and if $${\bf D}$$ admits a direct sum decomposition $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_1$$, $${\bf D}_2$$ satisfying the following conditions (1), (2B), and (3B): (1) $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$0\le i\le r-1$$, and $$k_{\frac{r-1}{2}}({\bf D}_1)=k_{\frac{r-1}{2}}({\bf D}_2)$$ is even. (2B) For both $${\bf D}={\bf D}_1$$ and $${\bf D}={\bf D}_2$$ we have $$\xi({\bf D})=\prod_{i=0}^{r-1}(k_i({\bf D})!)^{-1}$$ and $$k_i({\bf D})=k_{r-1-i}({\bf D})$$ for all $$1\le i\le \frac{r-1}{2}$$. (3B) $$s({\bf D}_2)-s({\bf D}_1)\equiv k_{0}({\bf D}_1)-k_{r-1}({\bf D}_1)$$ modulo $$(p-1)$$. (4) $$k_{\bullet}({\bf D}_1)\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Lemma 2.4. The conjunction of the conditions (1), (2C), and (3C) (respectively, (1), (2B), and (3B)) is symmetric in $${\bf D}_1$$ and $${\bf D}_2$$. □ Proof. That each one of the conditions (1), (2C), and (2B) is symmetric even individually is obvious. Now $$n({\bf D})\equiv0$$ modulo $$p-1$$ implies $$\sum_{i=0}^{r-1}k_{i}({\bf D}_1)\equiv0$$ modulo $$p-1$$. Therefore $$s({\bf D}_2)-s({\bf D}_1)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_1)$$ and $$k_{i}({\bf D}_1)=k_{r-1-i}({\bf D}_2)$$ for all $$i$$ (condition (1)) together imply $$s({\bf D}_1)-s({\bf D}_2)\equiv\sum_{i=0}^{r-1}ik_{i}({\bf D}_2)$$. Thus condition (3C) is symmetric, assuming condition (1). Similarly, condition (3B) is symmetric, assuming condition (1). ■ Definition. (i) Let $$\widetilde{\mathfrak S}_C(r)$$ denote the set of triples $$(n,s,\xi)$$ with integers $$1\le n\le p^r-2$$ and $$0\le s\le p-2$$ and scalars $$\xi\in k^{\times}$$ such that $$n\equiv 0$$ modulo $$(p-1)$$. Let $${\mathfrak S}_C(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_C(r)$$ by the involution   (∑i=0r−1kipi,s,ξ)↦(∑i=0r−1kr−i−1pi,s+∑i=0r−1iki,ξ). (Here and in the following, in the second component we mean the representative modulo $$p-1$$ belonging to $$[0,p-2]$$.) (ii) Let $$r$$ be odd and let $$\widetilde{\mathfrak S}_B(r)$$ denote the set of pairs $$(n,s)$$ with integers $$1\le n=\sum_{i=0}^{r-1}k_ip^i\le p^r-2$$ and $$0\le s\le p-2$$ such that $$n\equiv 0$$ modulo $$(p-1)$$ and such that $$k_i=k_{r-1-i}$$ for all $$1\le i\le \frac{r-1}{2}$$. Let $${\mathfrak S}_B(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_B(r)$$ by the involution   (∑i=0r−1kipi,s)↦(∑i=0r−1kr−i−1pi,s+k0−kr−1). □ Lemma 2.5. (i) Sending $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ to $$(n({\bf D}_1), s({\bf D}_1),\xi({\bf D}_1))$$ induces a bijection between the set of isomorphism classes of $$C$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_C(r)$$. (ii) Sending $${\bf D}={\bf D}_1\oplus{\bf D}_2$$ to $$(n({\bf D}_1), s({\bf D}_1))$$ induces a bijection between the set of isomorphism classes of $$B$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_B(r)$$. □ Proof. This follows from Lemma 2.3. ■ Definition. Let $$r$$ be even. Let $$\widetilde{\mathfrak S}_D(r)$$ denote the set of triples $$(n,s,\xi)$$ with integers $$1\le n=\sum_{i=0}^{r-1}k_ip^i\le p^r-2$$ and $$0\le s\le p-2$$ and scalars $$\xi\in k^{\times}$$ such that $$n\equiv 0$$ modulo $$(p-1)$$ and such that $$k_i=k_{i+\frac{r}{2}}$$ for all $$1\le i\le \frac{r}{2}-2$$. We consider the following permutations $$\iota_0$$ and $$\iota_1$$ of $$\widetilde{\mathfrak S}_D(r)$$. The value of $$\iota_0$$ at $$(\sum_{i=0}^{r-1}k_ip^i,s,\xi)$$ is   (kr2+∑i=1r2−2kipi+kr−1pr2−1+k0pr2+∑i=r2+1r−2kipi+kr2−1pr−1,s+∑i=0r2−1ki,ξ). The value of $$\iota_1$$ at $$(\sum_{i=0}^{r-1}k_ip^i,s,\xi)$$ is   (∑i=0r−1kr−i−1pi,s+r−24(kr2+k0)+∑i=2r2−1(i−1)kr2−i,ξ) if $$\frac{r}{2}$$ is odd, whereas if $$\frac{r}{2}$$ is even the value is   (kr2−1+∑i=1r2−2kr−i−1pi+kr2pr2−1+kr−1pr2+∑i=r2+1r−2kr−i−1pi+k0pr−1,s+(r4−1)kr2+r4k0+∑i=2r2−1(i−1)kr2−ipi,ξ). It is straightforward to check that $$\iota_0^2={\rm id}$$ and $$\iota_0\iota_1=\iota_1\iota_0$$, and moreover that $$\iota_1^2={\rm id}$$ if $$\frac{r}{2}$$ is odd, but $$\iota_1^2=\iota_0$$ if $$\frac{r}{2}$$ is even. In either case, the subgroup $$\langle\iota_0,\iota_1\rangle$$ of $${\rm Aut}({\widetilde{\mathfrak S}_D(r)})$$ generated by $$\iota_0$$ and $$\iota_1$$ is commutative and contains four elements. We let $${\mathfrak S}_D(r)$$ denote the quotient of $$\widetilde{\mathfrak S}_D(r)$$ by the action of $$\langle\iota_0,\iota_1\rangle$$. □ Definition. Let $$r$$ be even. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$D$$-symmetric if it admits a direct sum decomposition $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_{11}$$, $${\bf D}_{12}$$, $${\bf D}_{21}$$, $${\bf D}_{22}$$ satisfying the following conditions: (1) For all $$1\le i\le \frac{r}{2}-2$$ and all $$1\le s, t\le 2$$ we have $$k_i({\bf D}_{st})=k_{\frac{r}{2}+i}({\bf D}_{st})$$. (2) For all $$1\le i\le \frac{r}{2}-2$$ we have $$k_i({\bf D}_{11})=k_i({\bf D}_{12})$$ and $$k_i({\bf D}_{21})=k_i({\bf D}_{22})$$. (3) For $$j=1$$ and $$j=2$$ we have   k0(Dj1)=kr2(Dj2),kr2(Dj1)=k0(Dj2),kr2−1(Dj1)=kr−1(Dj2),kr−1(Dj1)=kr2−1(Dj2). (4)   ki(D11)=kr−i−1(D21) and ki(D12)=kr−i−1(D22) if $$i\in[0,r-1]$$ and $$\frac{r}{2}$$ is odd, or if $$i\in[1,\frac{r}{2}-2]\cup [\frac{r}{2}+1,r-2]$$ and $$\frac{r}{2}$$ is even. Moreover, if $$\frac{r}{2}$$ is even then   k0(D11)=kr−1(D21),kr2−1(D11)=k0(D21),kr2(D11)=kr2−1(D21),kr−1(D11)=kr2(D21). (5) $$\xi({\bf D}_{11})=\xi({\bf D}_{12})=\xi({\bf D}_{21})=\xi({\bf D}_{22})$$. (6) Modulo $$(p-1)$$ we have   s(Dj2)−s(Dj1)≡∑i=0r2−1ki(Dj1) for j=1,2s(D21)−s(D11)≡{r−24(kr2(D11)+k0(D11))+∑i=2r2−1(i−1)kr2−i(D11)r2 is odd(r4−1)kr2(D11)+r4k0(D11)+∑i=2r2−1(i−1)kr2−i(D11)r2 is even. (7) $$k_{\bullet}({\bf D}_{11})\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ Lemma 2.6. Sending $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ to $$(n({\bf D}_{11}), s({\bf D}_{11}),\xi({\bf D}_{11}))$$ induces a bijection between the set of isomorphism classes of $$D$$-symmetric étale $$(\varphi^r,\Gamma)$$-modules and $${\mathfrak S}_D(r)$$. □ Proof. Again we use Lemma 2.3. For a one-dimensional étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ put $$\alpha({\bf D})=(n({\bf D}), s({\bf D}),\xi({\bf D}))$$. If $${\bf D}={\bf D}_{11}\oplus{\bf D}_{12}\oplus{\bf D}_{21}\oplus{\bf D}_{22}$$ is $$D$$-symmetric as above, then $$\alpha({\bf D}_{st})$$ is an element of $$\widetilde{\mathfrak S}_D(r)$$, for all $$1\le s,t\le2$$. Moreover, it is straightforward to check $$\iota_0(\alpha({\bf D}_{11}))=\alpha({\bf D}_{12})$$, $$\iota_0(\alpha({\bf D}_{21}))=\alpha({\bf D}_{22})$$, $$\iota_1(\alpha({\bf D}_{11}))=\alpha({\bf D}_{21})$$, and $$\iota_1(\alpha({\bf D}_{12}))=\alpha({\bf D}_{22})$$. It follows that the above map is well-defined and bijective.■ Definition. We say that an étale $$(\varphi^r,\Gamma)$$-module $${\bf D}$$ over $$k_{{\mathcal E}}$$ is $$A$$-symmetric if $${\bf D}$$ admits a direct sum decomposition $${\bf D}=\oplus_{i=0}^{r-1}{\bf D}_i$$ with one-dimensional étale $$(\varphi^r,\Gamma)$$-modules $${\bf D}_i$$ satisfying   ki(Dj)=ki−j(D0),ξ(Dj)=ξ(D0),s(D0)−s(Dj)≡∑i=1jk−i(D0) modulo (p−1) for all $$i$$, $$j$$ (where we understand the sub index in $$k_{?}$$ as the unique representative in $$[0,r-1]$$ modulo $$r$$), and moreover   k∙(D0)∉{(0,…,0),(p−1,…,p−1)}. □ 3 Semi-infinite chamber galleries and the functor $${\bf D}$$ 3.1 Power multiplicative elements in the extended affine Weyl group Let $$G$$ be the group of $${\mathbb Q}_p$$-rational points of a $${\mathbb Q}_p$$-split connected reductive group over $${\mathbb Q}_p$$. Fix a maximal $${\mathbb Q}_p$$-split torus $$T$$ in $$G$$, let $$N(T)$$ be its normalizer in $$G$$. Let $$\Phi$$ denote the set of roots of $$T$$. For $$\alpha\in\Phi$$ let $$N_{\alpha}$$ be the corresponding root subgroup in $$G$$. Choose a positive system $$\Phi^+$$ in $$\Phi$$, let $$\Delta\subset\Phi^+$$ be the set of simple roots. Let $$N=\prod_{\alpha\in\Phi^+}N_{\alpha}$$. Let $$X$$ denote the semisimple Bruhat-Tits building of $$G$$, let $$A$$ denote its apartment corresponding to $$T$$. Our notational and terminological convention is that the facets of $$A$$ or $$X$$ are closed in $$X$$ (i.e., contain all their faces (the lower dimensional facets at their boundary)). A chamber is a facet of codimension $$0$$. For a chamber $$D$$ in $$A$$ let $$I_D$$ be the Iwahori subgroup in $$G$$ fixing $$D$$. We notice that $$I_D\cap N=\prod_{\alpha\in\Phi^+}I_{D}\cap N_{\alpha}$$. Fix a special vertex $$x_0$$ in $$A$$, let $$K$$ be the corresponding hyperspecial maximal compact open subgroup in $$G$$. Let $$T_0=T\cap K$$ and $$N_0=N\cap K$$. We have the isomorphism $$T/T_0\cong X_*(T)$$ sending $$\xi\in X_*(T)$$ to the class of $$\xi(p)\in T$$. Let $$I\subset K$$ be the Iwahori subgroup determined by $$\Phi^+$$. (If $${\rm red}:K\to\overline{K}$$ denotes the reduction map on to the reductive (over $${\mathbb F}_p)$$ quotient $$\overline{K}$$ of $$K$$, then $$I={\rm red}^{-1}({\rm red}(T_0N_0))$$.) Let $$C\subset A$$ be the chamber fixed by $$I$$. Thus $$I=I_C$$ and $$I\cap N=N_0=\prod_{\alpha\in\Phi^+}N_0\cap N_{\alpha}$$. We are interested in semi-infinite chamber galleries   C(0),C(1),C(2),C(3),… (3) in $$A$$ such that $$C=C^{(0)}$$ (and thus $$I=I_{C^{(0)}}$$) and such that, setting   N0(i)=IC(i)∩N, we have $$N_0=N_0^{(0)}$$ and   N0(0)⊃N0(1)⊃N0(2)⊃N0(3)⊃… with [N0(i):N0(i+1)]=p for all i≥0. (4) For any two chambers $$D_1\ne D_2$$ in $$A$$ sharing a common facet of codimension $$1$$ there is a uniquely determined $$\delta\in\Phi$$ with $$I_{D_1}\cap N_{\delta'}=I_{D_2}\cap N_{\delta'}$$ for $$\delta'\notin\{\delta,-\delta\}$$, with $$I_{D_1}\cap N_{-\delta}\subset I_{D_2}\cap N_{-\delta}$$ and with $$I_{D_2}\cap N_{\delta}\subset I_{D_1}\cap N_{\delta}$$, and both these inclusions are of index $$p$$. Moreover, the one-codimensional facet shared by $$D_1$$ and $$D_2$$ is contained in a translate of the hyperplane corresponding to $$\delta$$ (or equivalently, corresponding to $$-\delta$$).1 Applying this remark to $$D_1=C^{(i)}$$, $$D_2=C^{(i+1)}$$ and putting $$\alpha^{(i)}=\delta$$ we see that condition (4) precisely means that $$\alpha^{(i)}\in\Phi^+$$ for all $$i\ge0$$. Conversely, if condition (4) holds true then the sequence $$\alpha^{(0)},\alpha^{(1)},\alpha^{(2)},\alpha^{(3)},\ldots$$ in $$\Phi^+$$ can be characterized as follows: Setting   e[i,α]=|{0≤j≤i−1|α=α(j)}| for $$i\ge0$$ and $$\alpha\in\Phi^+$$, we have   N0(i)=∏α∈Φ+(N0∩Nα)pe[i,α]. Suppose that the center $$Z$$ of $$G$$ is connected. Then $$G/Z$$ is a semisimple group of adjoint type with maximal torus $$\check{T}=T/Z$$. Let $$\check{T}_0=T_0/(T_0\cap Z)\subset \check{T}$$. The extended affine Weyl group $$\widehat{W}=N(\check{T})/\check{T}_0$$ can be identified with the semidirect product between the finite Weyl group $$W=N(\check{T})/\check{T}=N(T)/T$$ and $$X_*(\check{T})$$. We identify $$A= X_*(\check{T})\otimes{\mathbb R}$$ such that $$x_0\in A$$ corresponds to the origin in the $${\mathbb R}$$-vector space $$X_*(\check{T})\otimes{\mathbb R}$$. We then regard $$\widehat{W}$$ as acting on $$A$$ through affine transformations. We regard $$\Delta\subset X^*({T})$$ as a subset of $$X^*(\check{T})$$. We usually enumerate the elements of $$\Delta$$ as $$\alpha_1,\ldots,\alpha_d$$, and we enumerate the corresponding simple reflection $$s_{\alpha}\in W$$ for $$\alpha\in\Delta$$ as $$s_1,\ldots,s_d$$ with $$s_i=s_{\alpha_i}$$. We assume that the root system $$\Phi$$ is irreducible. Let $${\alpha}_0\in\Phi$$ be the negative of the highest root. Let $$s_{\alpha_0}$$ be the corresponding reflection in the finite Weyl group $$W$$; define the affine reflection $$s_0=t_{\alpha_0^{\vee}}\circ s_{\alpha_0}\in \widehat{W}$$, where $$t_{\alpha_0^{\vee}}$$ denotes the translation by the coroot $$\alpha_0^{\vee}\in A$$ of $$\alpha_0$$. The affine Weyl group $$W_{{\rm aff}}$$ is the subgroup of $$\widehat{W}$$ generated by $$s_0,s_1,\ldots,s_d$$; in fact it is a Coxeter group with these Coxeter generators. The corresponding length function $$\ell$$ on $$W_{{\rm aff}}$$ extends to $$\widehat{W}$$. The above discussion shows that if the gallery (3) satisfies condition (4) and if $$w\in \widehat{W}$$ satisfies $$C^{(i)}=wC$$ for some $$i\ge0$$, then   pℓ(w)=pi=p[N0:IC(i)∩N0]. Let $$X_*(\check{T})_+$$ denote the set of dominant cocharacters. (Let $$T_+=\{t\in T\,|\,t N_0 t^{-1}\subset N_0\}$$, then $$X_*(\check{T})_+$$ is the image of $$T_+$$ under the map $$T_+\subset T\to T/T_0\cong X_*(T)\to X_*(\check{T})$$.) The monoid $$X_*(\check{T})_+$$ is free and has a distinguished basis $$\nabla$$, the set of fundamental coweights. The cone $${\mathcal C}$$ (vector chamber) in $$A$$ with origin in $$x_0$$ which is spanned by all the $$-\xi$$ for $$\xi\in \nabla$$ contains $$C$$, and $$C$$ is precisely the “top” chamber of this cone. The reflections $$s_0,s_1,\ldots,s_d$$ are precisely the reflections in the affine hyperplanes (walls) of $$A$$ which contain a codimension-$$1$$-face of $$C$$. Definition. We say that $$w\in \widehat{W}$$ is power multiplicative if $$\ell(w^m)=m\cdot\ell(w)$$ for all $$m\ge0$$. {(After finishing this article we found out that power multiplicative elements of $$\widehat{W}$$ have been studied, for example, in [3] under the name of straight elements.)} Of course, any element in the image of $$T\to N(\check{T})\to \widehat{W}=N(\check{T})/\check{T}_0$$ is power multiplicative because it acts on $$A$$ by translation (and $$\ell(w)$$ is the gallery distance between $$C$$ and $$wC$$). Suppose we are given a fundamental coweight $$\tau\in \nabla$$ and some non-trivial element $$\phi\in \widehat{W}$$ satisfying the following conditions: (a) $$\phi$$ is power multiplicative, (b) $$\tau$$ is minuscule, that is, we have $$\langle\alpha,\tau\rangle\in\{0,1\}$$ for all $$\alpha\in\Phi^+$$, (c) viewing $$\tau$$ via the embedding $$X_*(\check{T})\subset\widehat{W}$$ as an element of $$\widehat{W}$$, we have   ϕN∩τN≠∅. (5) □ Lemma 3.1. Let $$\phi$$ and $$\tau$$ be as above. Write $$\phi=\phi'v$$ with $$\phi'\in W_{{\rm aff}}$$ and $$v\in \widehat{W}$$ with $$vC=C$$. Choose a reduced expresssion   ϕ′=sβ(1)⋯sβ(r) of $$\phi'$$ with some function $$\beta:\{1,\ldots,r\}\to\{0,\ldots,d\}$$ (with $$r=\ell(\phi)=\ell(\phi')$$) and put   C(ar+b)=ϕasβ(1)⋯sβ(b)C (6) for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<r$$. Lift $$\tau\in \nabla\subset X_*(\check{T})$$ to some element of $$X_*({T})$$ and denote again by $$\tau$$ the corresponding homomorphism $${\mathbb Z}_p^{\times}\to T_0$$. Then we have: (i) The sequence   C=C(0),C(1),C(2),… satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (ii) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (iii) For any lifting $$\phi\in N(T)$$ of $$\phi\in \widehat{W}$$ we have $$\tau(a)\phi=\phi\tau(a)$$ in $$N(T)$$, for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. Given a chamber gallery (3) in $$A$$ with $$C=C^{(0)}$$ and $$N_0=N_0^{(0)}$$, condition (4) is equivalent with saying that (3) is a minimal chamber gallery in the cone $${\mathcal C}$$, that is, all $$C^{(i)}$$ are contained in $${\mathcal C}$$, and the gallery distance from $$C=C^{(0)}$$ to $$C^{(i)}$$ is $$i$$. Conversely, notice that if $$C^{(i)}$$ belongs to $${\mathcal C}$$ and the gallery distance from $$C=C^{(0)}$$ to $$C^{(i)}$$ is $$i$$, then also all $$C^{(j)}$$ for $$0\le j\le i$$ belong to $${\mathcal C}$$. (Otherwise the gallery would have to cross one of the bounding hyperplanes of $${\mathcal C}$$ and then cross it again later in the reverse direction; but a minimal gallery never a crosses a wall (hyperplane translate) back and forth.) Recall that, assuming condition (4), the set $$\{\alpha^{(j)}\,|\,j\ge0\}$$ is precisely the set of all $$\alpha\in\Phi^+$$ such that the gallery (3) crosses a translate of the hyperplane corresponding to $$\alpha$$. If there is some $$\epsilon\in X_*(T)$$ and some $$s\in{\mathbb N}$$ such that $$C^{(i+s)}=\epsilon C^{(i)}$$ for all $$i$$, then, since the action of $$\epsilon$$ is by translation on $$A$$, the latter set is the same as the set of all $$\alpha\in\Phi^+$$ such that the gallery $$C=C^{(0)},C^{(1)},\ldots, C^{(s)}$$ crosses a translate of the hyperplane corresponding to $$\alpha$$. Now let (3) be the chamber gallery defined by formula (6). (i) If $$n\in{\mathbb N}$$ is such that $$\phi^n\in X_*(T)$$, then the gallery distance between $$C$$ and $$\phi^nC$$ is $$\ell(\phi^n)$$. By construction, the gallery $$C=C^{(0)},C^{(1)},\ldots, C^{(n\ell(\phi))}=\phi^nC$$ has length $$n\ell(\phi)$$, that is, length $$\ell(\phi^n)$$ as $$\phi$$ is power multiplicative. Hence it must be a minimal chamber gallery. As $$\tau$$ is minuscule, it is in particular a dominant coweight. Therefore it follows from hypothesis (5) that also some power $$\phi^n$$ of $$\phi$$ is a dominant coweight. We thus obtain that $$C^{(n'\ell(\phi))}=\phi^{n'}C$$ lies in $${\mathcal C}$$ for each $$n'\in{\mathbb N}$$ divisible by $$n$$, but then, as the gallery is minimal and $${\mathcal C}$$ is a cone, $$C^{(i)}$$ lies in $${\mathcal C}$$ for each $$i\ge0$$. We get statement (i) in view of the preceeding discussion. (ii) The preceeding discussion shows $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,\langle\alpha,\phi^m\rangle\ne0\}$$ for any $$m\in{\mathbb N}$$ such that $$\phi^m$$ belongs to $$X_*(T)$$. As $$\phi$$ is power multiplicative, hypothesis (5) furthermore gives   {α(j)|j≥0}={α∈Φ+|⟨α,ϕm⟩≠0}={α∈Φ+|⟨α,τ⟩≠0} and as $$\tau$$ is minuscule this is the set   {α∈Φ+|⟨α,τ⟩=1}={α∈Φ+|α∘τ=idZp×}. (iii) By hypothesis (5) we have $$\tau^m=\phi^n$$ for some $$m,n\in{\mathbb N}$$. We deduce $$\tau^m=\phi\tau^m\phi^{-1}=(\phi\tau\phi^{-1})^{m}$$ and hence also $$\tau=\phi\tau\phi^{-1}$$ as $$\tau$$ and $$\phi\tau\phi^{-1}$$ belong to the free abelian group $$X_*(T)$$. Thus $$\tau\phi=\phi\tau$$ in $$\widehat{W}$$ which implies claim (iii). ■ Remark. For a given minuscule fundamental coweight $$\tau\in \nabla$$ any positive power $$\tau^m$$ of $$\tau$$ belongs to $$X_*(\check{T})$$, and so $$\phi=\tau^m$$ satisfies the assumptions of Lemma 3.1. However, for our purposes it is of interest to find $$\phi$$ (as in Lemma 3.1, possibly also required to project to $$W_{\rm aff}$$) of small length; the minimal positive power of $$\tau$$ belonging to $$X_*(\check{T})$$ is usually not optimal in this sense. □ 3.2 The functor D By $$I_0$$ we denote the pro-$$p$$-Iwahori subgroup contained in $$I$$. We often read $$\overline{T}=T_0/T_0\cap I_0$$ as a subgroup of $$T_0$$ by means of the Teichmüller character. Conversely, we read characters of $$\overline{T}$$ also as characters of $$T_0$$ (and do not introduce another name for these inflations). Let $${\rm ind}_{I_0}^{G}{\bf 1}_{{\mathfrak o}}$$ denote the $${\mathfrak o}$$-module of $${\mathfrak o}$$-valued compactly supported functions $$f$$ on $$G$$ such that $$f(ig)=f(g)$$ for all $$g\in G$$, all $$i\in I_0$$. It is a $$G$$-representation by means of $$(g'f)(g)=f(gg')$$ for $$g,g'\in G$$. Let   H(G,I0)=Endo[G](indI0G1o)op denote the corresponding pro-$$p$$-Iwahori Hecke algebra with coefficients in $${\mathfrak o}$$. For a subset $$H$$ of $$G$$ let $$\chi_H$$ denote the characteristic function of $$H$$. For $$g\in G$$ let $$T_g\in {\mathcal H}(G,I_0)$$ denote the Hecke operator corresponding to the double coset $$I_0gI_0$$. It sends $$f:G\to{\mathfrak o}$$ to   Tg(f):G⟶o,h↦∑x∈I0∖GχI0gI0(hx−1)f(x). Let $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ denote the category of $${\mathcal H}(G,I_0)$$-modules which as $${\mathfrak o}$$-modules are of finite length. We write $${\mathcal H}(G,I_0)_k={\mathcal H}(G,I_0)\otimes_{\mathfrak o}k$$. Given liftings $$\dot{s}\in N(T)$$ of all $$s\in S$$ we let $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ denote the $$k$$-subalgebra of $${\mathcal H}(G,I_0)_k$$ generated by the $$T_{\dot{s}}$$ for all $$s\in{S}$$ and the $$T_t$$ for $$t\in \overline{T}$$. Suppose we are given a reduced expression   ϕ=ϵs˙β(1)⋯s˙β(r) (7) (some function $$\beta:\{1,\ldots,r=\ell(\phi)\}\to\{0,\ldots,d\}$$, some $$\epsilon\in Z$$) of a power multiplicative element $$\phi\in N(T)$$, some power of which maps to a dominant coweight in $$N(T)/ZT_0$$. Put   C(ar+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<r$$. Then, by power multiplicativity of $$\phi$$, the sequence (3) thus defined satisfies property (4), cf. Lemma 3.1. Therefore we may use it to place ourselves into the setting (and notations) of [2], as follows. We define the half tree $$Y$$ whose edges are the elements in the $$N_0$$-orbits of the $$C^{(i)}\cap C^{(i+1)}$$ and whose vertices are the elements in the $$N_0$$-orbits of the $$C^{(i)}$$. We choose an isomorphism $$\Theta:Y\cong{\mathfrak X}_+$$ with the $$\lfloor{\mathfrak N}_0,\varphi,\Gamma\rfloor$$-equivariant half sub tree $${\mathfrak X}_+$$ of the Bruhat Tits tree of $${\rm GL}_2({\mathbb Q}_p)$$, satisfying the requirements of Theorem 3.2 of loc. cit. It sends the edge $$C^{(i)}\cap C^{(i+1)}$$ (respectively, the vertex $$C^{(i)}$$) of $$Y$$ to the edge $${\mathfrak e}_{i+1}$$ (respectively, the vertex $${\mathfrak v}_i$$) of $${\mathfrak X}_+$$. The half tree $$\overline{\mathfrak X}_+$$ is obtained from $${\mathfrak X}_+$$ by removing the “loose” edge $${\mathfrak e}_{0}$$. To an $${\mathcal H}(G,I_0)$$-module $$M$$ we associate the $$G$$-equivariant (partial) coefficient system $${\mathcal V}_M^X$$ on $$X$$. Briefly, its value at the chamber $$C$$ is $${\mathcal V}_M^X(C)=M$$. The transition maps $${\mathcal V}_M^X(D)\to{\mathcal V}_M^X(F)$$ for chambers (codimension-$$0$$-facets) $$D$$ and codimension-$$1$$-facets $$F$$ with $$F\subset D$$ are injective, and $${\mathcal V}_M^X(F)$$ for any such $$F$$ is the sum of the images of the $${\mathcal V}_M^X(D)\to{\mathcal V}_M^X(F)$$ for all $$D$$ with $$F\subset D$$. The pushforward $$\Theta_*{\mathcal V}_M$$ of the restriction of $${\mathcal V}_M^X$$ to $$Y$$ is a $$\lfloor{\mathfrak N}_0,\varphi^r,\Gamma_0\rfloor$$-equivariant coefficient system on $${\mathfrak X}_+$$ (and by further restriction on $$\overline{\mathfrak X}_+$$). This leads to the exact functor   M↦D(Θ∗VM)D(Θ∗VM)=H0(X¯+,Θ∗VM)⊗OE+OE=H0(X+,Θ∗VM)⊗OE+OE (8) from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{r},\Gamma_0)$$-modules over $${\mathcal O}_{\mathcal E}$$, where $$r=\ell(\phi)$$. If in addition we are given a homomorphism $$\tau:{\mathbb Z}_p^{\times}\to T_0$$ satisfying the conclusions of Lemma 3.1 (with respect to $$\phi$$), then this functor in fact takes values in the category of $$(\varphi^{r},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. For $$0\le i\le r-1$$ we put   yi=s˙β(1)⋯s˙β(i+1)s˙β(i)−1⋯s˙β(1)−1. Lemma 3.2. (a) $$y_i=y_{i-1}\cdots y_0\dot{s}_{\beta(i+1)}y_0^{-1}\cdots y_{i-1}^{-1}$$ and $$\phi=\epsilon y_{r-1}\cdots y_0$$. (b) $$y_i$$ is the affine reflection in the wall passing through $$C^{(i)}\cap C^{(i+1)}$$. □ Proof. To see (b) observe that $$y_i$$ indeed is a reflection, and that it sends $$C^{(i)}$$ to $$C^{(i+1)}$$. ■ 3.3 Supersingular modules For $$\alpha\in\Phi$$ we denote by $$\alpha^{\vee}$$ the associated coroot. For any $$\alpha\in\Phi$$ there is a corresponding homomorphism of algebraic groups $$\iota_{\alpha}:{\rm SL}_2({\mathbb Q}_p)\to G$$ as described in [5], Chapter II, Section 1.3. The element $$\iota_{\alpha}(\nu)$$ belongs to $$I\cap N_{\alpha}$$ and generates it as a topological group. For $$x\in{\mathbb F}_p^{\times}\subset {\mathbb Z}_p^{\times}$$ (via the Teichmüller character) we have $$\alpha^{\vee}(x)=\iota_{\alpha}(h(x))\in T$$. For $$0\le i\le d$$ let $$\overline{T}_{\alpha^{\vee}_i}$$ denote the subgroup of $$\overline{T}$$ given by the $${\mathbb F}_p$$-valued points of the schematic closure of $$\alpha^{\vee}_i(\overline{\mathbb F}_p^{\times})$$ (in the obvious $${\mathbb F}_p$$-group scheme (split torus) underlying $$\overline{T}$$). For a character $$\lambda:\overline{T}\to k^{\times}$$ let $$S_{\lambda}$$ be the subset of $$S=\{s_0,\ldots,s_d\}$$ consisting of all $$s_i$$ such that $$\lambda|_{\overline{T}_{\alpha^{\vee}_i}}$$ is trivial. Given $$\lambda$$ and a subset $${\mathcal J}$$ of $$S_{\lambda}$$ there is a uniquely determined character   χλ,J:H(G,I0)aff,k⟶k which sends $$T_t$$ to $$\lambda(t^{-1})$$ for $$t\in \overline{T}$$, which sends $$T_{\dot{s}}$$ to $$0$$ for $$s\in {S}-{\mathcal J}$$ and which sends $$T_{\dot{s}}$$ to $$-1$$ for $$s\in {\mathcal J}$$ (see [8] Proposition 2). For $$0\le i\le d$$ we define a number $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ such that   λ(αi∨(x))=xkiuad for allx∈Fp×, (9) as follows. If $$s_i\in{\mathcal J}$$ put $$k_i=p-1$$. Otherwise let $$k_i$$ be the unique integer in $$[0,p-2]$$ satisfying formula (9). Vignéras defined the notion of a supersingular$${\mathcal H}(G,I_0)_k$$-module, see [7, 8]. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-subalgebra of $${\mathcal H}(G,I_0)_k$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with all $$T_z$$ for $$z\in Z$$. Theorem 3.3. Vignéras, Ollivier) (a) Let $$\chi:{\mathcal H}(G,I_0)'_{{\rm aff},k}\to k$$ be a character extending $$\chi_{\lambda,{\mathcal J}}:{\mathcal H}(G,I_0)_{{\rm aff},k}\longrightarrow k$$ for some $$\lambda$$, $${\mathcal J}$$. If $$k_{\bullet}\notin\{(0,\ldots,0),(p-1,\ldots,p-1)\}$$ then the $${\mathcal H}(G,I_0)_k$$-module $${\mathcal H}(G,I_0)_k\otimes_{{\mathcal H}(G,I_0)'_{{\rm aff},k}}\chi$$ is supersingular. (b) Any supersingular $${\mathcal H}(G,I_0)_k$$-module of finite length contains a character $$\chi_{\lambda,{\mathcal J}}:{\mathcal H}(G,I_0)_{{\rm aff},k}\longrightarrow k$$ for some $$\lambda$$, $${\mathcal J}$$ with $$k_{\bullet}\notin\{(0,\ldots,0),(p-1,\ldots,p-1)\}$$. □ Proof. See [7] Theorem 5.14, Corollary 5.16, at least for the case where the root system underlying $$G$$ is irreducible. See [9] for more general statements. ■ It follows that the simple supersingular $${\mathcal H}(G,I_0)_k$$-modules are precisely the simple quotients of the $${\mathcal H}(G,I_0)_k$$-modules $${\mathcal H}(G,I_0)_k\otimes_{{\mathcal H}(G,I_0)'_{{\rm aff},k}}\chi$$ appearing in statement (a) of Theorem 3.3. We call the latter ones standard supersingular$${\mathcal H}(G,I_0)_k$$-modules. In fact, it is easy to see that these are simple themselves if the pair $$(k_{\bullet},{\mathcal J})$$ satisfies a certain non-periodicity property. 4 Classical matrix groups For $$m\in{\mathbb N}$$ let $$E_m\in{\rm GL}_m$$ denote the identity matrix and let $${E^*_d}$$ denote the standard antidiagonal element in $${\rm GL}_d$$ (i.e., the permutation matrix of maximal length). Let   S^m=(Em−Em),Sm=(EmEm). For $$1\le i,j\le m$$ with $$i\ne j$$ let $$\epsilon_{i,j}\in {\rm GL}_m$$ denote the matrix with entries $$1$$ on the diagonal and at the spot $$(i,j)$$ and with entries $$0$$ otherwise. 4.1 Affine root system $$\boldsymbol{\tilde{C}_d}$$} Assume $$d\ge2$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (s0s1)4=(sd−1sd)4=1 and (si−1si)3=1 for 2≤i≤d−1 (10) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and usiu=sd−i for 0≤i≤d. (11) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Consider the symplectic and the general symplectic group   Sp2d(Qp)={A∈GL2d(Qp)|TAS^dA=S^d},G=GSp2d(Qp)={A∈GL2d(Qp)|TAS^dA=κ(A)S^d for some κ(A)∈Qp×}. Let $$T$$ denote the maximal torus in $$G$$ consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T∩Sp2d(Qp)⟶Qp×,diag(x1,…,x2d)↦xi. For $$1\le i,j\le d$$ and $$\epsilon_1, \epsilon_2\in\{\pm 1\}$$ we thus obtain characters (using additive notation as usual) $$\epsilon_1e_i+\epsilon_2e_j:T\cap {\rm Sp}_{2d}({\mathbb Q}_p)\longrightarrow {\mathbb Q}_p^{\times}$$. We extend these latter ones to $$T$$ by setting   ϵ1ei+ϵ2ej:T⟶Qp×,A=diag(x1,…,x2d)↦xiϵ1xjϵ2κ(A)−ϵ1−ϵ22. For $$i=j$$ and $$\epsilon=\epsilon_1=\epsilon_2$$ we simply write $$\epsilon 2 e_i$$. Then $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}\cup\{\pm 2e_i\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$C_d$$. We choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}\cup\{2e_i\,|\,1\le i\le d\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=2e_d\}$$. The negative of the highest root is $$\alpha_0=-2e_1$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,Ed−1,x,Ed−1)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1)1≤i≤d−1diag(Ed−1,x,Ed−1,x−1)i=d. (12) For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm GL}_{2d}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S^1,Ed−i−1,Ei−1,S^1,Ed−i−1) and furthermore   s˙d=(Ed−11Ed−1−1),s˙0=(−p−1Ed−1pEd−1). Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $${\rm Sp}_{2d}({\mathbb Q}_p)\subset G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/ZT_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (19). Put   u˙=(Ed∗pEd∗). Then $$\dot{u}$$ belongs to $$N(T)$$ and normalizes $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/ZT_0$$ satisfies the relations (20). In $$N(T)$$ we consider   ϕ=(p⋅id)s˙ds˙d−1⋯s˙1s˙0. We may rewrite this as $$\phi=(p\cdot{\rm id})\dot{s}_{\beta(1)}\cdots\dot{s}_{\beta(d+1)}$$ where we put $$\beta(i)=d+1-i$$ for $$1\le i\le d+1$$. For $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<d+1$$ we put   C(a(d+1)+b)=ϕasd⋯sd−b+1C=ϕasβ(1)⋯sβ(b)C. Define the homomorphism   τ:Zp×⟶T0,x↦diag(xEd,Ed). Lemma 4.1. We have $$\phi^{d}\in T$$ and $$\phi^{d}N_0\phi^{-d}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For all $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) This is Lemma 3.1 in practice. A matrix computation shows $$\phi^{d}=(-1)^{d-1}{\rm diag}(p^{d+1}E_{d},p^{d-1}E_{d})\in T$$. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, by $$\epsilon_{i,i+d}$$ if $$\alpha=2e_i$$ with $$1\le i\le d$$, and by $$\epsilon_{i,j}\epsilon^{-1}_{i+d,j+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$. Using this we find   ϕdN0ϕ−d=∏α∈Φ+ϕd(N0∩Nα)ϕ−d=∏α∈Φ+(N0∩Nα)pmα,  mα={2α=ei+ej with 1≤i<j≤d2α=2ei with 1≤i≤d0all other α∈Φ+. In particular we find $$\phi^{d}N_0\phi^{-d}\subset N_0$$ and $$[N_0:\phi^{d}N_0\phi^{-d}]=p^{d(d+1)}$$. Let $$n\in {\mathbb N}$$. We have $$\ell(\phi^{n})\le{n(d+1)}$$ because the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$d+1$$ Coxeter generators. If $$m\ge n$$ is such that $$m\in d{\mathbb N}$$ we then have, on the other hand,   pℓ(ϕm)≥[I0:(I0∩ϕmIϕ−m)]≥[N0:(N0∩ϕmI0ϕ−m)]=[N0:ϕmN0ϕ−m]=pm(d+1). Thus   ℓ(ϕn)≥ℓ(ϕm)−ℓ(ϕm−n)≥m(d+1)−(m−n)(d+1)=n(d+1), hence $$\ell(\phi^{n})={n(d+1)}$$. We have shown that $$\phi$$ is power multiplicative and   [N0:ϕdN0ϕ−d]=[N0:(N0∩ϕdN0ϕ−d)]=[I0:(I0∩ϕdI0ϕ−d)]. We get $$I_0=N_0\cdot (I_0\cap\phi^{d}I_0\phi^{-d})$$ and that hypothesis (4) holds true. (b) As $$\phi^d\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation. ■ As explained in Section 3.2 we now obtain a functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{d+1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. As explained in [2], to compute it we need to understand the intermediate objects $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, acted on by $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma\rfloor$$. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{u}^{2}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ (see the proof of [8] Proposition 3). Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$2$$, a $$k$$-basis is $$e,f$$ where we write $$e=1\otimes e$$ and $$f=T_{\dot{u}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$0\le j\le d$$ put $$\widetilde{j}=d-j$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=(p⋅id)s˙β~(1)⋯s˙β~(d+1). Put $$n_e=\sum_{i=0}^{d}k_{d-i}p^i=\sum_{i=0}^{d}k_{\beta(i+1)}p^i$$ and $$n_f=\sum_{i=0}^{d}k_{i}p^i=\sum_{i=0}^{d}k_{\widetilde{\beta}(i+1)}p^i$$. Put $$\varrho=\prod_{i=0}^{d}(k_{i}!)=\prod_{i=0}^d(k_{\beta(i+1)}!)=\prod_{i=0}^d(k_{\widetilde{\beta}(i+1)}!)$$. Let $$0\le s_e, s_f\le p-2$$ be such that $$\lambda(\tau(x))=x^{-s_e}$$ and $$\lambda(\dot{u}\tau(x)\dot{u}^{-1})=x^{-s_f}$$ for all $$x\in{\mathbb F}_p^{\times}$$. Lemma 4.2. The assignment $$M[\lambda,{\mathcal J},b]\mapsto (n_e,s_e,b\varrho^{-1})$$ induces a bijection between the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_C(d+1)$$. □ Proof. We have $$\prod_{i=0}^d\alpha^{\vee}_{i}(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (12)). This implies   ∑i=0dki≡ne≡nf≡0 mod (p−1). (13) One can deduce from [8] Proposition 3 that for two sets of data $$\lambda, {\mathcal J}, b$$ and $$\lambda', {\mathcal J}', b'$$ the $${\mathcal H}(G,I_0)_k$$-modules $$M[\lambda,{\mathcal J},b]$$ and $$M[\lambda',{\mathcal J}',b']$$ are isomorphic if and only if $$b=b'$$ and the pair $$(\lambda, {\mathcal J})$$ is conjugate with the pair $$(\lambda', {\mathcal J}')$$ by means of a power of $$\dot{u}$$, that is, by means of $$\dot{u}^0=1$$ or $$\dot{u}^1=\dot{u}$$. Conjugating $$(\lambda,{\mathcal J})$$ by $$\dot{u}$$ has the effect of substituting $$k_{d-i}$$ with $$k_i$$, for any $$i$$. The datum of the character $$\lambda$$ is equivalent with the datum of $$s_e$$ together with all the $$k_i$$ taken modulo $$(p-1)$$ since the images of $$\tau$$ and all $$\alpha_i^{\vee}$$ together generate $$\overline{T}$$. Knowing the set $${\mathcal J}$$ is then equivalent with knowing the numbers $$k_i$$ themselves (not just modulo $$(p-1)$$). Thus, our mapping is well-defined and bijective. ■ Fix $$M=M[\lambda,{\mathcal J},b]$$. Let $$0\le j\le d$$. Normalize the homomorphism $$\iota_{\alpha_j}:{\rm SL}_2({\mathbb Q}_p)\to G$$ such that $$\iota_{\alpha_j}(\dot{s})=\dot{s}_j$$ for $$\dot{s}=\left( {0 \atop 1}\quad {{-1} \atop {0}}\right)$$. Let $$t_j=\iota_{\alpha_j}([\nu])-1\in k[[\iota_{\alpha_j}{\mathfrak N}_0]]\subset k[[N_0]]$$. Let $$F_j$$ denote the codimension-$$1$$-face of $$C$$ contained in the (affine) reflection hyperplane (in $$A\subset X$$) for $$s_j$$. Lemma 4.3. In $${\mathcal V}_M^X(F_j)$$ we have $$t_j^{k_j}\dot{s}_je=k_j!e$$ and $$t_j^{k_{d-j}}\dot{s}_jf=k_{d-j}!f$$ for all $$0\le j\le d$$. □ Proof. Let $$I_0^{j}$$ denote the subgroup of $$G$$ generated by $$I_0$$ and $$\dot{s}_{j}I_0 \dot{s}_{j}^{-1}$$. Let $$\overline{I}_0^{j}$$ denote the maximal reductive (over $${\mathbb F}_p$$) quotient of $$I_0^{j}$$. Then $$\iota_{\alpha_j}({\rm SL}_2({\mathbb Z}_p))\subset I_0^{j}$$ and $$\iota_{\alpha_j}$$ induces an isomorphism between either the quotient $$\overline{\mathcal S}={\rm SL}_2({\mathbb F}_p)$$ of $${\rm SL}_2({\mathbb Z}_p)$$, or the quotient $$\overline{\mathcal S}={\rm PSL}_2({\mathbb F}_p)$$ of $${\rm SL}_2({\mathbb Z}_p)$$, with the quotient $$\overline{I}_0^{j}$$ of $$I_0^{j}$$. The subgroup $$\overline{\mathcal U}$$ of $$\overline{\mathcal S}$$ generated by (the image of) $$\nu$$ maps isomorphically to the image of $$I_0$$ in $$\overline{I}_0^{j}$$. Therefore we get an embedding of $$k$$-algebras   H(S¯,U¯)k=Endk[S¯](indU¯S¯1k)op≅Endk[I0j](indI0I0j1k)op↪H(G,I0)aff,k⊂H(G,I0)k. Let $$\rho:{\mathcal H}(\overline{\mathcal S},\overline{\mathcal U})_k\to k$$ be the character defined by $$\rho(T_{\dot{s}})=-1$$ if $$k_{j}=p-1$$ but $$\rho(T_{\dot{s}})=0$$ if $$0\le k_{j}<p-1$$, and by $$\rho(T_{h(x)})=x^{-k_{j}}$$. We have a morphism of $${\rm SL}_2({\mathbb Z}_p)$$-representations (acting on $${\mathcal V}_M^X(F_j)$$ through $$\iota_{\alpha_j}$$)   (indU¯S¯1k)⊗H(S¯,U¯)kρ⟶VMX(Fj) sending a basis element $$e$$ of $$\rho$$ to $$e\in{\mathcal V}_M^X(F_j)$$. It is injective since its restriction to the space $$k.e$$ of $${\mathcal U}$$-invariants (see [2] Lemma 2.3) is injective. By [2] Lemma 2.5 we have $$t^{k_{j}}\dot{s}e=k_{j}!e$$ in $$({\rm ind}_{\overline{\mathcal U}}^{\overline{\mathcal S}}{\bf 1}_k)\otimes_{{\mathcal H}(\overline{\mathcal S},\overline{\mathcal U})_k}\rho$$ where $$t=[\nu]-1\in k[\overline{\mathcal U}]$$. We get $$t_j^{k_j}\dot{s}_je=k_j!e$$ in $${\mathcal V}_M^X(F_j)$$. A similar argument gives $$t_j^{k_{d-j}}\dot{s}_jf=k_{d-j}!f$$. For this notice that as the Hecke operator $$T_t$$ for $$t\in\overline{T}$$ acts on $$k.e$$ through $$\lambda(t^{-1})$$, it acts on $$k.f=k.T_{\dot{u}}e$$ through $$\lambda (\dot{u}t^{-1}\dot{u}^{-1})$$ (the same computation as in formula (18) below), and that formula (9) implies $$\lambda (\dot{u}\alpha_j^{\vee}(x)\dot{u}^{-1})=x^{k_{d-j}}$$. ■ Lemma 4.4. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tneφd+1e=ϱb−1e, (14)  tnfφd+1f=ϱb−1f, (15)  γ(x)e=x−see, (16)  γ(x)f=x−sff (17) for $$x\in{\mathbb F}_p^{\times}$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. The construction of $$\Theta$$ in [2] is based on the choice of elements $$\nu_i\in N_0$$ topologically generating the relevant pro-$$p$$ subgroup “at distance $$i$$’, for each $$0\le i\le d$$. These $$\nu_i$$ can be taken to be the elements $$y_{i-1}\cdots y_0\iota_{d-i}(\nu)y_0^{-1}\cdots y_{i-1}^{-1}$$. Their actions on the halftree $$Y$$ correspond to the actions of the $$\nu^{p^{i}}$$ on the halftree $${\mathfrak X}_+$$. We use the notations and the statements of Lemma 3.2, observing $$\beta(i+1)=d-i$$. For $$0\le i\le d$$ we have $$y_{i-1}\cdots y_0F_{d-i}=C^{(i)}\cap C^{(i+1)}$$ and $$y_{i-1}\cdots y_0C=C^{(i)}$$. Thus $$y_{i-1}\cdots y_0$$ defines isomorphisms   VMX(Fd−i)≅VMX(C(i)∩C(i+1)) and VMX(C)≅VMX(C(i)). But we have   VMX(C(i)∩C(i+1))=Θ∗VM(vi) and VMX(C(i))=Θ∗VM(ei). It follows that, under the above isomorphisms, the action of $$t_{d-i}$$, respectively, of $$\dot{s}_{d-i}$$, on $${\mathcal V}_M^X(F_{d-i})$$ becomes the action of $$[\nu]^{p^{i}}-1$$, respectively, of $$y_i$$, on $$\Theta_*{\mathcal V}_M({\mathfrak v}_i)$$. Now as we are in characteristic $$p$$ we have $$t^{p^i}=([\nu]-1)^{p^{i}}=[\nu]^{p^{i}}-1$$. Applying this to the element $$e$$, respectively, $$f$$, of $${\mathcal V}_M^X(C)\subset{\mathcal V}_M^X(F_{d-i})$$, Lemma 4.3 tells us   (tpi)kd−iyi⋯y0e=kd−i!yi−1⋯y0e, respectively, (tpi)kiyi⋯y0f=ki!yi−1⋯y0f. We compose these formulae for all $$0\le i\le d$$ and finally recall that the central element $$p\cdot{\rm id}$$ acts on $$M$$ through the Hecke operator $$T_{p^{-1}\cdot{\rm id}}$$, that is, by $$b^{-1}$$. We get formulae (15) and (14). Next recall that the action of $$\gamma(x)$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is given by that of $$\tau(x)\in T$$, that is, by the Hecke operator $$T_{\tau(x)^{-1}}$$. We thus compute   γ(x)e=Tτ(x)−1e=λ(τ(x))e,γ(x)f=Tτ(x)−1Tu˙e=Tu˙τ(x)−1e=Tu˙Tu˙τ(x)−1u˙−1e=Tu˙λ(u˙τ(x)u˙−1)e=λ(u˙τ(x)u˙−1)f (18) and obtain formulae (16) and (17).■ Corollary 4.5. The étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_e$$, $$g_f$$ such that   φd+1ge=bϱ−1tne+1−pd+1geφd+1gf=bϱ−1tnf+1−pd+1gfγ(x)ge−xsege∈t⋅kE+⋅geγ(x)gf−xsfgf∈t⋅kE+⋅gf. □ Proof. This follows from Lemma 4.4 as explained in [2] Lemma 6.4.■ Corollary 4.6. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. For $$x\in{\mathbb F}_p^{\times}$$ we have   τ(x)⋅u˙τ−1(x)u˙−1=diag(xEd,x−1Ed)=(∑i=0d(i+1)αi∨)(x) in $$\overline{T}$$. Applying $$\lambda$$ and observing $$\sum_{i=0}^dk_i\equiv 0$$ modulo $$(p-1)$$ we get   xsf−se=λ((∑i=0d(i+1)αi∨)(x))=x∑i=0diki and hence $$s_f-s_e\equiv \sum_{i=0}^dik_i$$ modulo $$(p-1)$$. Therefore   D(Θ∗VM)=D(Θ∗VM)1⊕D(Θ∗VM)2 with $${\bf D}(\Theta_*{\mathcal V}_M)_1=k_{\mathcal E}g_e$$ and $${\bf D}(\Theta_*{\mathcal V}_M)_2=k_{\mathcal E}g_f$$ is a direct sum decomposition of $${\bf D}(\Theta_*{\mathcal V}_M)$$ as an étale $$(\varphi^{d+1},\Gamma)$$-module over $$k_{\mathcal E}$$, and by Corollary (4.5) this decomposition identifies $${\bf D}(\Theta_*{\mathcal V}_M)$$ as being $$C$$-symmetric (recall in particular that $$n_e=\sum_{i=0}^{d}k_{d-i}p^i$$ and $$n_f=\sum_{i=0}^{d}k_{i}p^i$$). Composing this assignment $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ (from standard supersingular $${\mathcal H}(G,I_0)_k$$-modules to $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules) with the bijection of Lemma 2.5 (i) (from $$C$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules to $${\mathfrak S}_C(d+1)$$) we get the bijection of Lemma 4.2 (from standard supersingular $${\mathcal H}(G,I_0)_k$$-modules to $${\mathfrak S}_C(d+1)$$). Thus, $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ is a bijection as stated.■ Remark. Consider the subgroup $$G'={\rm Sp}_{2d}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d},x^{-1}E_{d})$$ and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d}\dot{s}_{d-1}\cdots \dot{s}_{1}\dot{s}_0$$ then everything in fact happens inside $$G'$$. We then have $$\alpha^{(j)}\circ\tau={\rm id}^2_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which $$\tau(-1)$$ (i.e., $$T_{\tau(-1)}=T_{\tau(-1)^{-1}}$$) acts trivially. For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma^2\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^2=\{\gamma^2\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, following [2] (as a slight variation from what we explained in Section 3.2), we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{d+1},\Gamma^2)$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. In the case $$d=2$$ one may also work with $$\phi=(p\cdot{\rm id})\dot{s}_2\dot{s}_1\dot{s}_2\dot{u}$$. Its square is $$p\cdot{\rm id}$$ times the square of the $$\phi=(p\cdot{\rm id})\dot{s}_2\dot{s}_1\dot{s}_0$$ used above. □ 4.2 Affine root system $$\boldsymbol{\tilde{B}_d}$$ Assume $$d\ge 3$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (sd−1sd)4=1 and (s0s2)3=(si−1si)3=1 for 2≤i≤d−1 (19) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us0u=s1 and usiu=si for 2≤i≤d. (20) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Let   S~d=(Sd001)∈GL2d+1(Qp) and consider the special orthogonal group   G=SO2d+1(Qp)={A∈SL2d+1(Qp)|TAS~dA=S~d}. Let $$T$$ denote the maximal torus consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T⟶Qp×,diag(x1,…,xd,x1−1,…,xd−1,1)↦xi. Then (in additive notation) $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}\cup\{\pm e_i\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$B_d$$. We choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}\cup\{e_i\,|\,1\le i\le d\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=e_d\}$$. The negative of the highest root is $$\alpha_0=-e_1-e_2$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,x−1,Ed−2,x,x,Ed−2,1)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1,1)1≤i≤d−1diag(Ed−1,x2,Ed−1,x−2,1)i=d. (21) Remark. For roots $$\alpha\in\Phi$$ of the form $$\alpha=\pm e_i\pm e_j$$ the homomorphism $$\iota_{\alpha}:{\rm SL}_2\to {\rm SO}_{2d+1}$$ is injective. For roots $$\alpha\in\Phi$$ of the form $$\alpha=\pm e_i$$ the homomorphism $$\iota_{\alpha}:{\rm SL}_2\to {\rm SO}_{2d+1}$$ induces an embedding $${\rm PSL}_2\to {\rm SO}_{2d+1}$$. □ For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm SL}_{2d+1}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S1,Ed−i−1,Ei−1,S1,Ed−i−1,1) and furthermore   s˙d=(Ed−11Ed−11−1). Define   u˙=(p−1Ed−1pEd−1−1) and $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}$$. Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $$G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/T_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (19). Also $$\dot{u}$$ belongs to $$G$$; it normalizes $$T$$, $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/T_0=\widehat{W}$$ satisfies the relations (20). In $$N(T)$$ we consider the element   ϕ=s˙1s˙2⋯s˙d−1s˙ds˙d−1⋯s˙2s˙0. (22) We may rewrite this as $$\phi=\dot{s}_{\beta(1)}\cdots\dot{s}_{\beta(2d-1)}$$ where we put $$\beta(i)=i$$ for $$1\le i\le d$$ and $$\beta(i)=2d-i$$ for $$d\le i\le 2d-2$$ and $$\beta(2d-1)=0$$. We put   C(a(2d−1)+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<2d-1$$. Define the homomorphism   τ:Zp×⟶T0,x↦diag(x,Ed−1,x−1,Ed−1,1). Lemma 4.7. We have $$\phi^{2}\in T$$ and $$\phi^{2}N_0\phi^{-2}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) A matrix computation shows $$\phi^{2}={\rm diag}(p^2,E_{d-1},p^{-2},E_{d-1},1)$$. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon^{-1}_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, by $$\epsilon_{i,j}\epsilon^{-1}_{j+d,i+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$, and by $$\epsilon_{i,2d+1}\epsilon^{-1}_{2d+1,i+d}$$ if $$\alpha=e_i$$ with $$1\le i\le d$$. Using this we find   ϕ2N0ϕ−2=∏α∈Φ+ϕ2(N0∩Nα)ϕ−2=∏α∈Φ+(N0∩Nα)pmα,mα={2α=e1−ei with 1<i2α=e1+ei with 1<i2α=e10all other α∈Φ+. In particular we find $$\phi^{2}N_0\phi^{-2}\subset N_0$$ and $$[N_0:\phi^{2}N_0\phi^{-2}]=p^{2(2d-1)}$$. On the other hand, the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$2d-1$$ Coxeter generators. Arguing as in the proof of Lemma 4.1 we combine these facts to obtain $$\ell(\phi)=2d-1$$ and that $$\phi$$ is power multiplicative. We obtain that hypothesis (4) holds true, again by the same reasoning as in Lemma 4.1. (b) As $$\phi^{2}\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{2d-1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$ and a subset $${\mathcal J}\subset S_{\lambda}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. Notice that $$k_d$$ is necessarily even since in $$\alpha_d^{\vee}(x)$$ (formula (21)) the entry $$x$$ only appears squared. Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J]=H(G,I0)k⊗H(G,I0)aff,kk.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J}}$$. As a $$k$$-vector space, $$M$$ has dimension $$2$$, a $$k$$-basis is $$e,f$$ where we write $$e=1\otimes e$$ and $$f=T_{\dot{u}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J}]$$ for some $$\lambda, {\mathcal J}$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. A packet of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules is a set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules all of which give rise to the same $${\mathcal J}$$ and the same $$k_{\bullet}=(k_i)_{i}$$. □ Remark. Let $$\overline{T}'$$ denote the subgroup of $$\overline{T}$$ generated by the $$\alpha_i^{\vee}({\mathbb F}_p^{\times})$$ for all $$0\le i\le d$$. Formula (21) implies $$[\overline{T}:\overline{T}']=2$$. Two supersingular $${\mathcal H}(G,I_0)_k$$-modules $$M[\lambda,{\mathcal J}]$$ and $$M[\lambda',{\mathcal J}']$$ belong to the same packet if and only if $${\mathcal J}={\mathcal J}'$$ and if the restrictions of $$\lambda$$ and $$\lambda'$$ to $$\overline{T}'$$ coincide. □ For $$2\le j\le d$$ we put $$\widetilde{j}=j$$, furthermore we put $$\widetilde{0}=1$$ and $$\widetilde{1}=0$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=s˙β~(1)⋯s˙β~(2d−1). Put $$n_e=\sum_{i=0}^{2d-2}k_{\beta(i+1)}p^i$$ and $$n_f=\sum_{i=0}^{2d-2}k_{\widetilde{\beta}(i+1)}p^i$$. Put $$\varrho=k_0!k_1!k_d!\prod_{i=2}^{d-1}(k_{i}!)^2=\prod_{i=0}^{2d-2}(k_{\beta(i+1)}!)=\prod_{i=0}^{2d-2}(k_{\widetilde{\beta}(i+1)}!)$$. Let $$0\le s_e, s_f\le p-2$$ be such that $$\lambda(\tau(x))=x^{-s_e}$$ and $$\lambda(\dot{u}\tau(x)\dot{u}^{-1})=x^{-s_f}$$ for all $$x\in{\mathbb F}_p^{\times}$$. Lemma 4.8. The assignment $$M[\lambda,{\mathcal J}]\mapsto (n_e,s_e)$$ induces a bijection between the set of packets of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_B(2d-1)$$. □ Proof. We have $$\alpha^{\vee}_{0}(x)\alpha^{\vee}_{1}(x)\alpha^{\vee}_{{d}}(x)\prod_{i=2}^{d-1}(\alpha_{i}^{\vee})^2(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (21)). This implies   k0+k1+kd+2∑i=2d−1ki≡ne≡nf≡0 mod (p−1). (23) We further proceed exactly as in the proof of Lemma 4.2.■ Lemma 4.9. Let $$M=M[\lambda,{\mathcal J}]$$ for some $$\lambda$$, $${\mathcal J}$$. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tneφ2d−1e=ϱe, (24)  tnfφ2d−1f=ϱf, (25)  γ(x)e=x−see, (26)  γ(x)f=x−sff (27) for $$x\in{\mathbb F}_p^{\times}$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. As in Lemma 4.4. ■ Corollary 4.10. The étale $$(\varphi^{2d-1},\Gamma)$$-module over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_e$$, $$g_f$$ such that   φ2d−1ge=ϱ−1tne+1−p2d−1geφ2d−1gf=ϱ−1tnf+1−p2d−1gfγ(x)ge−xsege∈t⋅kE+⋅geγ(x)gf−xsfgf∈t⋅kE+⋅gf. □ Proof. This follows from Lemma 4.9 as explained in [2] Lemma 6.4.■ Corollary 4.11. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of packets of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$B$$-symmetric étale $$(\varphi^{2d-1},\Gamma)$$-modules $${\bf D}$$ over $$k_{\mathcal E}$$. □ Proof. For $$x\in{\mathbb F}_p^{\times}$$ we compute   τ(x)⋅u˙τ−1(x)u˙−1=diag(x2,Ed−1,x−2,Ed−1,1)=(α1∨−α0∨)(x) in $$\overline{T}$$. Application of $$\lambda$$ gives $$x^{s_f-s_e}=x^{k_1-k_0}$$ and hence $$s_f-s_e\equiv k_1-k_0=k_{\beta(1)}-k_{\beta(2d-1)}$$ modulo $$(p-1)$$. The required symmetry in the $$p$$-adic digits of $$n_e$$, $$n_f$$ is due to the corresponding symmetry of the function $$\beta$$. Thus, $${\bf D}(\Theta_*{\mathcal V}_M)$$ is a $$B$$-symmetric étale $$(\varphi^{2d-1},\Gamma)$$-module. One checks that composing $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ with the map of Lemma 2.5 (ii) gives the map of Lemma 4.8. As the maps in Lemmata 4.8 and 2.5 (ii) are bijective we obtain our result.■ 4.3 Affine root system $$\boldsymbol{\tilde{D}_d}$$ Assume $$d\ge4$$. Here $$W_{{\rm aff}}$$ is the Coxeter group with generators $$s_0, s_1,\ldots,s_d$$ and relations   (sd−2sd)3=(s0s2)3=(si−1si)3=1 for 2≤i≤d−1 (28) and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us0u=s1,us1u=s0,usd−1u=sd,usdu=sd−1,usiu=si for 2≤i≤d−2. (29) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with a four-element subgroup $$W_{\Omega}$$, with $$u\in W_{\Omega}$$ of order $$2$$.) Consider the general orthogonal group   GO2d(Qp)={A∈GL2d(Qp)|TASdA=κ(A)Sd for some κ(A)∈Qp×}. It contains the special orthogonal group   SO2d(Qp)={A∈SL2d(Qp)|TASdA=Sd}. Let $$G={\rm GSO}_{2d}({\mathbb Q}_p)$$ be the subgroup of $${\rm GO}_{2d}({\mathbb Q}_p)$$ generated by $${\rm SO}_{2d}({\mathbb Q}_p)$$ and by all $${\rm diag}(xE_d,E_d)$$ with $$x\in{\mathbb Q}_p^{\times}$$; it is of index $$2$$ in $${\rm GO}_{2d}({\mathbb Q}_p)$$. {(With the obvious definitions, $${\rm GO}_{2d}({\mathbb Q}_p)$$ is the group of $${\mathbb Q}_p$$-valued points of an algebraic group $${\rm GO}_{2d}$$, whereas $$G$$ is the group of $${\mathbb Q}_p$$-valued points of the connected component $${\rm GSO}_{2d}$$ of $${\rm GO}_{2d}$$; this $${\rm GSO}_{2d}$$ has connected center.) Let $$T$$ be the maximal torus consisting of all diagonal matrices in $$G$$. For $$1\le i\le d$$ let   ei:T∩SL2d(Qp)⟶Qp×,diag(x1,…,x2d)↦xi. For $$1\le i,j\le d$$ and $$\epsilon_1, \epsilon_2\in\{\pm 1\}$$ we thus obtain characters (using additive notation as usual) $$\epsilon_1e_i+\epsilon_2e_j:T\cap {\rm SL}_{2d}({\mathbb Q}_p)\longrightarrow {\mathbb Q}_p^{\times}$$. We extend these latter ones to $$T$$ by setting   ϵ1ei+ϵ2ej:T⟶Qp×,A=diag(x1,…,x2d)↦xiϵ1xjϵ2κ(A)−ϵ1−ϵ22. Then $$\Phi=\{\pm e_i\pm e_j\,|\,i\ne j\}$$ is the root system of $$G$$ with respect to $$T$$. It is of type $$D_d$$. Choose the positive system $$\Phi^+=\{ e_i\pm e_j\,|\,i< j\}$$ with corresponding set of simple roots $$\Delta=\{\alpha_1=e_1-e_2, \alpha_2=e_2-e_3,\ldots,\alpha_{d-1}=e_{d-1}-e_d, \alpha_{d}=e_{d-1}+e_d\}$$. The negative of the highest root is $$\alpha_0=-e_1-e_2$$. For $$0\le i\le d$$ we have the following explicit formula for $$\alpha_i^{\vee}=(\alpha_i)^{\vee}$$:   αi∨(x)={diag(x−1,x−1,Ed−2,x,x,Ed−2)i=0diag(Ei−1,x,x−1,Ed−i−1,Ei−1,x−1,x,Ed−i−1)1≤i≤d−1diag(Ed−2,x,x,Ed−2,x−1,x−1)i=d. (30) For $$\alpha\in \Phi$$ let $$N_{\alpha}^0$$ be the subgroup of the corresponding root subgroup $$N_{\alpha}$$ of $$G$$ all of whose elements belong to $${\rm SL}_{2d}({\mathbb Z}_p)$$. Let $$I_0$$ denote the pro-$$p$$-Iwahori subgroup generated by the $$N_{\alpha}^0$$ for all $$\alpha\in \Phi^+$$, by the $$(N_{\alpha}^0)^p$$ for all $$\alpha \in \Phi^-=\Phi-\Phi^+$$, and by the maximal pro-$$p$$-subgroup of $$T_0$$. Let $$I$$ denote the Iwahori subgroup of $$G$$ containing $$I_0$$. Let $$N_0$$ be the subgroup of $$G$$ generated by all $$N_{\alpha}^0$$ for $$\alpha\in \Phi^+$$. For $$1\le i\le d-1$$ define the block diagonal matrix   s˙i=diag(Ei−1,S1,Ed−i−1,Ei−1,S1,Ed−i−1)=diag(Ei−1,S1,Ed−2,S1,Ed−i−1). Put   u˙=(p−1Ed−21pEd−21) and $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}$$ and $$\dot{s}_d=\dot{u}\dot{s}_{d-1}\dot{u}$$. Then $$\dot{s}_{i}$$ for $$0\le i\le d$$ belongs to $${\rm SO}_{2d}({\mathbb Q}_p)\subset G$$ and normalizes $$T$$. Its image element $$s_{i}=s_{\alpha_i}$$ in $$N(T)/ZT_0=\widehat{W}$$ is the reflection corresponding to $$\alpha_i$$. The $${s}_0,{s}_1,\ldots,{s}_{d-1},{s}_{d}$$ are Coxeter generators of $$W_{\rm aff}\subset \widehat{W}$$ satisfying the relations (28). Also $$\dot{u}$$ belongs to $${\rm SO}_{2d}({\mathbb Q}_p)\subset G$$; it normalizes $$T$$, $$I$$ and $$I_0$$. Its image element $$u$$ in $$N(T)/ZT_0=\widehat{W}$$ satisfies the relations (29). In $$N(T)$$ we consider   ϕ=(p⋅id)s˙d−1s˙d−2⋯s˙2s˙1s˙ds˙d−2s˙d−3⋯s˙3s˙2s˙0 if d iseven,ϕ=(p2⋅id)s˙d−1s˙d−2⋯s˙2s˙1s˙ds˙d−2s˙d−3⋯s˙3s˙2s˙0 if d isodd. (The reason for our distinction according to the parity of $$d$$ is that the center $$Z$$ of $$G$$ is the subgroup of $$G$$ generated by $$T_0\cap Z$$ and $$p\cdot{\rm id}$$ if $$d$$ is even, respectively, by $$T_0\cap Z$$ and $$p^2\cdot{\rm id}$$ if $$d$$ is odd.) We may rewrite this as $$\phi=(p\cdot{\rm id})\dot{s}_{\beta(1)}\cdots \dot{s}_{\beta(2d-2)}$$ if $$d$$ is even, respectively, $$\phi=(p^2\cdot{\rm id})\dot{s}_{\beta(1)}\cdots \dot{s}_{\beta(2d-2)}$$ if $$d$$ is odd, where $$\beta(i)=d-i$$ for $$1\le i\le d-1$$, $$\beta(d)=d$$, $$\beta(i)=2d-1-i$$ for $$d+1\le i\le 2d-3$$ and $$\beta(2d-2)=0$$. We put   C(a(2d−2)+b)=ϕasβ(1)⋯sβ(b)C for $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<2d-2$$. Define the homomorphism   τ:Zp×⟶T0,x↦diag(xEd−1,Ed,x). Lemma 4.12. We have $$\phi^{d}\in T$$ and $$\phi^{d}N_0\phi^{-d}\subset N_0$$. The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). In particular we may define $$\alpha^{(j)}\in\Phi^+$$ for all $$j\ge0$$. (b) For any $$j\ge0$$ we have $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$. (c) We have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. □ Proof. (a) A matrix computation shows $$\phi^{d}={\rm diag}(p^{d+2}E_{d-1},p^{d-2}E_{d},p^{d+2})$$ if $$d$$ is even, and $$\phi^{d}={\rm diag}(p^{2d+2}E_{d-1},p^{2d-2}E_{d},p^{2d+2})$$ if $$d$$ is odd. The group $$N_{\alpha}$$ for $$\alpha\in\Phi^+$$ is generated by $$\epsilon_{i,j+d}\epsilon^{-1}_{j,i+d}$$ if $$\alpha=e_i+e_j$$ with $$1\le i<j\le d$$, and by $$\epsilon_{i,j}\epsilon^{-1}_{j+d,i+d}$$ if $$\alpha=e_i-e_j$$ with $$1\le i<j\le d$$. Using this we find   ϕdN0ϕ−d=∏α∈Φ+ϕd(N0∩Nα)ϕ−d=∏α∈Φ+(N0∩Nα)pmα,mα={4α=ei+ej with 1≤i<j<d4α=ei−ed with 1≤i<d0all other α∈Φ+. In particular we find $$\phi^{d}N_0\phi^{-d}\subset N_0$$ and $$[N_0:\phi^{d}N_0\phi^{-d}]=p^{2d(d-1)}$$. On the other hand, the image of $$\phi$$ in $$\widehat{W}$$ is a product of $$2d-2$$ Coxeter generators and of an element of length $$0$$. Arguing as in the proof of Lemma 4.1 we combine these facts to obtain $$\ell(\phi)=2d-2$$ and that $$\phi$$ is power multiplicative. We obtain that hypothesis (4) holds true, again by the same reasoning as in Lemma 4.1. (b) As $$\phi^{d}\in T$$ we have $$\{\alpha^{(j)}\,|\,j\ge0\}=\{\alpha\in\Phi^+\,|\,m_{\alpha}\ne0\}$$. This implies (b). (c) Another matrix computation.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{2d-2},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Consider the elements   ω˙=(Ed∗pEd∗),ρ˙=(Ed−1∗ppEd−1∗1) of $${\rm GO}_{2d}({\mathbb Q}_p)$$. They normalize $$T$$ and satisfy   ω˙u˙=u˙ω˙,ω˙2=p⋅idω˙s˙iω˙−1=s˙d−i for 0≤i≤d,ρ˙2=p⋅u˙,ρ˙s˙iρ˙−1=s˙d−i for 2≤i≤d−2,ρ˙s˙d−1ρ˙−1=s˙1,ρ˙s˙dρ˙−1=s˙0,ρ˙s˙0ρ˙−1=s˙d−1,ρ˙s˙1ρ˙−1=s˙d. The element $$\dot{\omega}$$ belongs to $$G$$ if and only if $$d$$ is even. The element $$\dot{\rho}$$ belongs to $$G$$ if and only if $$d$$ is odd. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{\omega}^{2}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$ if $$d$$ is even, respectively, $$T_{p^2\cdot {\rm id}}=T_{\dot{\rho}^{4}}$$ and $$T^{-1}_{p^2\cdot {\rm id}}=T_{p^{-2}\cdot {\rm id}}$$ if $$d$$ is odd. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ if $$d$$ is even, respectively, which sends $$T_{p^2\cdot {\rm id}}$$ to $$b$$ if $$d$$ is odd (see the proof of [8] Proposition 3). We define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$4$$. A $$k$$-basis is $$e_0,e_1,f_0,f_1$$ where we write   e0=1⊗e,f0=Tu˙⊗e,e1=Tω˙⊗e,f1=Tu˙ω˙⊗e if d is even,e0=1⊗e,f0=Tu˙⊗e,e1=Tρ˙⊗e,f1=Tu˙ρ˙⊗e if d is odd. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$2\le j\le d-2$$ let $$\widetilde{j}=j$$, and furthermore let $$\widetilde{d-1}=d$$ and $$\widetilde{d}=d-1$$ and $$\widetilde{1}=0$$ and $$\widetilde{0}=1$$. Letting $$\widetilde{\beta}=\widetilde{(.)}\circ\beta$$ we then have   u˙ϕu˙−1=(pn⋅id)s˙β~(1)⋯s˙β~(2d−2) with $$n=1$$ if $$d$$ is even, but $$n=2$$ if $$d$$ is odd. If $$d$$ is odd we consider in addition the following two maps $$\gamma$$ and $$\delta$$ from $$[1,2d-2]$$ to $$[0,d]$$. We put $$\gamma(1)=1$$, $$\gamma(d-1)=d$$, $$\gamma(d)=0$$, and $$\gamma(2d-2)=d-1$$. We put $$\delta(1)=0$$, $$\delta(d-1)=d-1$$, $$\delta(d)=1$$, and $$\delta(2d-2)=d$$. We put $$\gamma(i)=\delta(i)=\beta(2d-2-i)$$ for all $$i\in[2,\ldots,d-2]\cup[d+1,\ldots,2d-3]$$. We then have   ϱ˙ϕϱ˙−1=(p2⋅id)s˙γ(1)⋯s˙γ(2d−2),ϱ˙−1ϕϱ˙=(p2⋅id)s˙δ(1)⋯s˙δ(2d−2). Fix $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$. Put   ne0=∑i=02d−3kβ(i+1)pi,nf0=∑i=02d−3kβ~(i+1)pi forany parity of d,ne1=∑i=02d−3kβ(2d−2−i)pi,nf1=∑i=02d−3kβ~(2d−2−i)pi if d is even,ne1=∑i=02d−3kγ(i+1)pi,nf1=∑i=02d−3kδ(i+1)pi if d is odd. Let $$0\le s_{e_0}, s_{f_0}, s_{e_1}, s_{f_1}\le p-2$$ be such that for all $$x\in{\mathbb F}_p^{\times}$$ we have   λ(τ(x))=x−se0,λ(u˙τ(x)u˙−1)=x−sf0 forany parity of d,λ(ω˙τ(x)ω˙−1)=x−se1,λ(ω˙u˙τ(x)u˙−1ω˙−1)=x−sf1 if d is even,λ(ρ˙τ(x)ρ˙−1)=x−se1,λ(ρ˙u˙τ(x)u˙−1ρ˙−1)=x−sf1 if d is odd. Put $$\varrho=k_0!k_1!k_{d-1}!k_d!\prod_{i=2}^{d-2}(k_i!)^2=\prod_{i=0}^{2d-3}(k_{\beta(i+1)}!)=\prod_{i=0}^{2d-3}(k_{\widetilde{\beta}(i+1)}!)$$. Lemma 4.13. The assignment $$M[\lambda,{\mathcal J},b]\mapsto (n_{e_0}, s_{e_0}, b\varrho^{-1})$$ induces a bijection between the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules and $${\mathfrak S}_D(2d-2)$$. □ Proof. We have $$\alpha^{\vee}_{0}(x)\alpha^{\vee}_{1}(x)\alpha^{\vee}_{{d-1}}(x)\alpha^{\vee}_d(x)\prod_{i=2}^{d-2}(\alpha_{i}^{\vee})^2(x)=1$$ for all $$x\in{\mathbb F}_p^{\times}$$ (as can be seen e.g., from formula (30)). This implies   k0+k1+kd−1+kd+2∑i=2d−2ki≡ne0≡nf0≡ne1≡nf1≡0 mod (p−1). (31) It follows from [8] Proposition 3 that $$M[\lambda,{\mathcal J},b]$$ and $$M[\lambda',{\mathcal J}',b']$$ are isomorphic if and only if $$b=b'$$ and the pair $$(\lambda, {\mathcal J})$$ is conjugate with the pair $$(\lambda', {\mathcal J}')$$ by means of $$\dot{u}^{n}\dot{\omega}^m$$ for some $$n,m\in\{0,1\}$$ (if $$d$$ is even), respectively, by means of $$\dot{u}^{n}\dot{\rho}^m$$ for some $$n,m\in\{0,1\}$$ (if $$d$$ is odd). Under the map $$M[\lambda,{\mathcal J},b]\mapsto (n_{e_0}, s_{e_0}, b\varrho^{-1})$$, conjugation by $$\dot{u}$$ corresponds to the permutation $$\iota_0$$ of $$\widetilde{\mathfrak S}_D(2d-2)$$, while conjugation by $$\dot{\omega}$$, respectively, by $$\dot{\rho}$$, corresponds to the permutation $$\iota_1$$ of $$\widetilde{\mathfrak S}_D(2d-2)$$. We may thus proceed as in the proof of Lemma 4.2 to see that our mapping is well-defined and bijective.■ Lemma 4.14. For an appropriate choice of the isomorphism $$\Theta$$ we have, in $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$,   tnejφ2d−2ej=ϱb−1ej,tnfjφ2d−2fj=ϱb−1fj,γ(x)ej=x−sejej,γ(x)fj=x−sfjfj for $$x\in{\mathbb F}_p^{\times}$$ and $$j=0,1$$. The action of $$\Gamma_0$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ is trivial on the subspace $$M$$. □ Proof. As in Lemma 4.4.■ Corollary 4.15. The étale $$(\varphi^{2d-2},\Gamma)$$-module over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$g_{e_0}$$, $$g_{f_0}$$, $$g_{e_1}$$, $$g_{f_1}$$ such that for both $$j=0$$ and $$j=1$$ we have   φ2d−2gej=bϱ−1tnej+1−p2d−2gejφ2d−2gfj=bϱ−1tnfj+1−p2d−2gfjγ(x)(gej)−xsejgej∈t⋅kE+⋅gejγ(x)(gfj)−xsfjgfj∈t⋅kE+⋅gfj. □ Proof. This follows from Lemma 4.14 as explained in [2] Lemma 6.4.■ Corollary 4.16. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$D$$-symmetric étale $$(\varphi^{2d-2},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. We first verify that $${\bf D}(\Theta_*{\mathcal V}_M)$$ for $$M=M[\lambda,{\mathcal J},b]$$ is indeed $$D$$-symmetric. For this let $${\bf D}_{11}=\langle g_{e_0}\rangle$$, $${\bf D}_{12}=\langle g_{f_0}\rangle$$, $${\bf D}_{21}=\langle g_{e_1}\rangle$$, $${\bf D}_{22}=\langle g_{f_1}\rangle$$. Then $$k_i({\bf D}_{11})=k_{\beta(i+1)}$$ and $$k_i({\bf D}_{12})=k_{\widetilde{\beta}(i+1)}$$; moreover $$k_i({\bf D}_{21})=k_{\beta(2d-d-i)}$$ and $$k_i({\bf D}_{22})=k_{\widetilde{\beta}(2d-d-i)}$$ if $$d$$ is even, but $$k_i({\bf D}_{21})=k_{\gamma(i+1)}$$ and $$k_i({\bf D}_{22})=k_{\delta(i+1)}$$ if $$d$$ is odd. For the condition on $$s_{f_0}-s_{e_0}=s({\bf D}_{12})-s({\bf D}_{11})$$ we compute   τ(x)⋅u˙τ−1(x)u˙−1=diag(x,Ed−2,x−1,x−1,Ed−2,x)=(∑i=1d−1αi∨)(x), hence application of $$\lambda$$ gives $$x^{s_{f_0}-s_{e_0}}=x^{\sum_{i=1}^{d-1}k_i}$$ and hence $$s_{f_0}-s_{e_0}\equiv \sum_{i=1}^{d-1}k_i=\sum_{i=0}^{d-2}k_i({\bf D}_{11})$$ modulo $$(p-1)$$. The condition on $$s_{f_1}-s_{e_1}=s({\bf D}_{22})-s({\bf D}_{21})$$ in case $$d$$ is even is exactly verified like the one for $$s_{f_0}-s_{e_0}$$ because $$\dot{\omega}\tau(x)\dot{\omega}^{-1}\cdot\dot{\omega}\dot{u}\tau^{-1}(x)\dot{u}^{-1}\dot{\omega}^{-1}=\tau(x)\cdot\dot{u}\tau^{-1}(x)\dot{u}^{-1}$$. In case $$d$$ is odd the computation is   ρ˙τ(x)ρ˙−1⋅ρ˙u˙τ−1(x)u˙−1ρ˙−1=diag(x,Ed−2,x,x−1,Ed−2,x−1)=(αd∨+∑i=1d−2αi∨)(x), hence $$s_{f_1}-s_{e_1}\equiv k_d+\sum_{i=1}^{d-2}k_i=s({\bf D}_{12})-s({\bf D}_{11})+k_d-k_{d-1}=\sum_{i=0}^{d-2}k_i({\bf D}_{21})$$ modulo $$(p-1)$$. To see the condition on $$s_{e_1}-s_{e_0}=s({\bf D}_{21})-s({\bf D}_{11})$$ in case $$d$$ is even we compute   τ(x)⋅ω˙τ−1(x)ω˙−1=diag(1,xEd−2,1,1,x−1Ed−2,1)=(d−22αd−1∨+d−22αd∨+∑i=2d−2(i−1)αi∨)(x), (32) hence application of $$\lambda$$ gives $$x^{s_{e_1}-s_{e_0}}=x^{\frac{d-2}{2}k_{d-1}+\frac{d-2}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i}$$ and hence $$s_{e_1}-s_{e_0}\equiv \frac{d-2}{2}k_{d-1}+\frac{d-2}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i=\frac{d-2}{2}(k_{d-1}({\bf D}_{11})+k_{0}({\bf D}_{11}))+\sum_{i=2}^{d-2}(i-1)k_{d-i-1}({\bf D}_{11})$$ modulo $$(p-1)$$. If however $$d$$ is odd we compute   τ(x)⋅ρ˙τ−1(x)ρ˙−1=diag(1,xEd−2,x−1,1,x−1Ed−2,x)=(d−12αd−1∨+d−32αd∨+∑i=2d−2(i−1)αi∨)(x), hence application of $$\lambda$$ gives $$x^{s_{e_1}-s_{e_0}}=x^{\frac{d-1}{2}k_{d-1}+\frac{d-3}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i}$$ and hence $$s_{e_1}-s_{e_0}\equiv \frac{d-1}{2}k_{d-1}+\frac{d-3}{2}k_{d}+\sum_{i=2}^{d-2}(i-1)k_i=\frac{d-3}{2}k_{\frac{r}{2}}({\bf D}_{11})+\frac{d-1}{2}k_{0}({\bf D}_{11})+\sum_{i=2}^{d-2}(i-1)k_{d-i-1}({\bf D}_{11})$$ modulo $$(p-1)$$. We have shown that $${\bf D}(\Theta_*{\mathcal V}_M)$$ is $$D$$-symmetric. One checks that composing $$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$$ with the map of Lemma 2.6 gives the map of Lemma 4.13. As the maps in Lemmata 4.13 and 2.6 are bijective we obtain our result.■ Remark. Consider the subgroup $$G'={\rm SO}_{2d}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d-1},x^{-1}E_{d},x)$$, and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d-1}\dot{s}_{d-2}\cdots \dot{s}_2\dot{s}_1\dot{s}_{d}\dot{s}_{d-2}\dot{s}_{d-3}\cdots \dot{s}_3\dot{s}_2\dot{s}_0$$, then everything in fact happens inside $$G'$$, and there is no dichotomy between $$d$$ even or odd. We then have $$\alpha^{(j)}\circ\tau={\rm id}^2_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which $$\tau(-1)$$ (i.e., $$T_{\tau(-1)}=T_{\tau(-1)^{-1}}$$) acts trivially. For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{2d-2},\Gamma^2\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^2=\{\gamma^2\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{2d-2},\Gamma^2)$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. Instead of the element $$\phi\in N(T)$$ used above we might also work with the element $$\dot{s}_{d-1}\cdots \dot{s}_2\dot{s}_1\dot{u}$$ of length $$d-1$$ (or products of this with elements of $$p^{\mathbb Z}\cdot{\rm id}$$), keeping the same $$C^{(\bullet)}$$. This results in a functor from $${\mathcal H}(G,I_0)$$-modules to $$(\varphi^{d-1},\Gamma)$$-modules. Up to a factor in $$p^{\mathbb Z}\cdot{\rm id}$$, the square of $$\dot{s}_{d-1}\cdots \dot{s}_2\dot{s}_1\dot{u}$$ is the element $$\phi$$ used above. □ Remark. For the affine root system of type $$D_d$$ there are three co minuscule fundamental coweights (cf. [1] Chapter 8, par 7.3]). The corresponding $$\phi$$’s for the other two choices are longer. □ 4.4 Affine root system $$\tilde{A}_d$$ Assume $$d\ge1$$ and consider $$G={\rm GL}_{d+1}({\mathbb Q}_p)$$. Let   u˙=(Edp). For $$1\le i\le d$$ let   s˙i=diag(Ei−1,S1,Ed−i) and let $$\dot{s}_0=\dot{u}\dot{s}_1\dot{u}^{-1}$$. Let $$T$$ be the maximal torus consisting of diagonal matrices. Let $$\Phi^+$$ be such that $$N=\prod_{\alpha\in\Phi^+}N_{\alpha}$$ is the subgroup of upper triangular unipotent matrices. Let $$I_0$$ be the subgroup consisting of elements in $${\rm GL}_{d+1}({\mathbb Z}_p)$$ which are upper triangular modulo $$p$$. We put   ϕ=(p⋅id)s˙d⋯s˙0=(p⋅id)s˙β(1)⋯s˙β(d+1), where $$\beta(i)=d+1-i$$ for $$1\le i\le d+1$$. For $$a,b\in{\mathbb Z}_{\ge0}$$ with $$0\le b<d+1$$ we put   C(a(d+1)+b)=ϕasd⋯sd−b+1C=ϕasβ(1)⋯sβ(b)C. We define the homomorphism   τ:Zp×⟶T0,x↦diag(Ed,x−1). The sequence $$C=C^{(0)}, C^{(1)}, C^{(2)},\ldots$$ satisfies hypothesis (4). The corresponding $$\alpha^{(j)}\in\Phi^+$$ for $$j\ge0$$ satisfy $$\alpha^{(j)}\circ\tau={\rm id}_{{\mathbb Z}_p^{\times}}$$, and we have $$\tau(a)\phi=\phi\tau(a)$$ for all $$a\in{\mathbb Z}_p^{\times}$$. We thus obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{d+1},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Let $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ denote the $$k$$-sub algebra of $${\mathcal H}(G,I_0)_{k}$$ generated by $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ together with $$T_{p\cdot {\rm id}}=T_{\dot{u}^{d+1}}$$ and $$T^{-1}_{p\cdot {\rm id}}=T_{p^{-1}\cdot {\rm id}}$$. Suppose we are given a character $$\lambda:\overline{T}\to k^{\times}$$, a subset $${\mathcal J}\subset S_{\lambda}$$ and some $$b\in k^{\times}$$. Define the numbers $$0\le k_i=k_i(\lambda,{\mathcal J})\le p-1$$ as in Section 3.3. The character $$\chi_{\lambda,{\mathcal J}}$$ of $${\mathcal H}(G,I_0)_{{\rm aff},k}$$ extends uniquely to a character   χλ,J,b:H(G,I0)aff,k′⟶k which sends $$T_{p\cdot {\rm id}}$$ to $$b$$ (see the proof of [8] Proposition 3). Define the $${\mathcal H}(G,I_0)_k$$-module   M=M[λ,J,b]=H(G,I0)k⊗H(G,I0)aff,k′k.e, where $$k.e$$ denotes the one-dimensional $$k$$-vector space on the basis element $$e$$, endowed with the action of $${\mathcal H}(G,I_0)'_{{\rm aff},k}$$ by the character $$\chi_{\lambda,{\mathcal J},b}$$. As a $$k$$-vector space, $$M$$ has dimension $$d+1$$, a $$k$$-basis is $$\{e_i\}_{0\le i\le d}$$ where we write $$e_i=T_{\dot{u}^{-i}}\otimes e$$. Definition. We call an $${\mathcal H}(G,I_0)_k$$-module standard supersingular if it is isomorphic with $$M[\lambda,{\mathcal J},b]$$ for some $$\lambda, {\mathcal J}, b$$ such that $$k_{\bullet}\notin\{(0,\ldots,0), (p-1,\ldots,p-1)\}$$. □ For $$0\le j\le d$$ put $$n_{e_j}=\sum_{i=0}^dk_{j-i}p^i$$ (reading $$j-i$$ as its representative modulo $$(d+1)$$ in $$[0,d]$$) and let $$s_{e_j}$$ be such that $$\lambda(\dot{u}^{-j}\tau(x)\dot{u}^j)=x^{-s_{e_j}}$$. Put $$\varrho=\lambda(-{\rm id})\prod_{i=0}^d(k_i!)$$. Theorem 4.17. The étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ over $$k_{\mathcal E}$$ associated with $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$ admits a $$k_{\mathcal E}$$-basis $$\{g_{e_j}\}_{0\le j\le d}$$ such that for all $$j$$ we have   φd+1gej=bϱ−1tnej+1−pd+1gej,γ(x)gej−xsejgej∈t⋅kE+⋅gej. The functor $$M\mapsto {\bf D}(\Theta_*{\mathcal V}_M)$$ induces a bijection between (a) the set of isomorphism classes of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules, and (b) the set of isomorphism classes of $$A$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-modules over $$k_{\mathcal E}$$. □ Proof. For the formulae describing $${\bf D}(\Theta_*{\mathcal V}_M)$$ one may proceed exactly as in the proof of Corollary 4.5. (The only tiny additional point to be observed is that the $$\dot{s}_i$$ (in keeping with our choice in [2]) do not lay in the images of the $$\iota_{\alpha_i}$$; this is accounted for by the sign factor $$\lambda(-{\rm id})$$ in the definition of $$\varrho$$.) Alternatively, as our $$\phi$$ is the $$(d+1)$$st power of the $$\phi$$ considered in Section 8 of [2], the computations of loc. cit. may be carried over. To see that $${\bf D}(\Theta_*{\mathcal V}_M)$$ is $$A$$-symmetric put $${\bf D}_j=\langle g_{e_j}\rangle$$ for $$0\le j\le d$$ and compare the above formulae with those defining $$A$$-symmetry; for example, we find $$s_{e_0}-s_{e_j}\equiv\sum_{i=1}^jk_i$$ modulo $$(p-1)$$. The bijectivity statement is then verified as before.■ Remark. Application of the functor of Lemma 2.1 to any one of the direct summands $${\bf D}_j$$ of the $$A$$-symmetric étale $$(\varphi^{d+1},\Gamma)$$-module $${\bf D}(\Theta_*{\mathcal V}_M)$$ yields an étale $$(\varphi,\Gamma)$$-module isomorphic with the one assigned to $$M$$ in [2]. □ Remark. Consider the subgroup $$G'={\rm SL}_{d+1}({\mathbb Q}_p)$$ of $$G$$. If we replace the above $$\tau$$ by $$\tau:{\mathbb Z}_p^{\times}\longrightarrow T_0$$, $$x\mapsto {\rm diag}(xE_{d},x^{-d})$$ and if we replace the above $$\phi$$ by $$\phi=\dot{s}_{d}\dot{s}_{d-1}\cdots \dot{s}_{1}\dot{s}_0$$ then everything in fact happens inside $$G'$$. We then have $$\alpha^{(j)}\circ\tau={\rm id}^{d+1}_{{\mathbb Z}_p^{\times}}$$ for all $$j\ge0$$. Let $${\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ denote the category of finite-$${\mathfrak o}$$-length $${\mathcal H}(G',G'\cap I_0)$$-modules on which the $$xE_{d+1}$$ (i.e., the $$T_{x^{-1}E_{d+1}}$$) for all $$x\in{\mathbb Z}_p^{\times}$$ with $$x^{d+1}=1$$ act trivially. (Notice that $$\tau(x)=xE_{d+1}$$ for such $$x$$.) For $$M\in{\rm Mod}_0^{\rm fin}{\mathcal H}(G',G'\cap I_0)$$ we obtain an action of $$\lfloor{\mathfrak N}_0,\varphi^{d+1},\Gamma^{d+1}\rfloor$$ on $$H_0(\overline{\mathfrak X}_+,\Theta_*{\mathcal V}_M)$$, where $$\Gamma^{d+1}=\{\gamma^{d+1}\,|\,\gamma\in\Gamma\}\subset\Gamma$$. Correspondingly, we obtain a functor from $${\rm Mod}^{\rm fin}_0{\mathcal H}(G',G'\cap I_0)$$ to the category of $$(\varphi^{d+1},\Gamma^{d+1})$$-modules over $${\mathcal O}_{\mathcal E}$$. □ Remark. As all the fundamental coweights $$\tau$$ of $$T$$ are minuscule, each of them admits $$\phi$$’s for which the pair $$(\phi,\tau)$$ satisfies the properties asked for in Lemma 3.1. For example, let $$1\le g\le d$$. For $$\phi=\dot{s}_g\cdot \dot{s}_{g+1}\cdots \dot{s}_d\cdot \dot{u}$$ as well as for $$\phi=\dot{s}_g\cdot \dot{s}_{g-1}\cdots \dot{s}_1\cdot \dot{u}^{-1}$$ there is a unique minimal gallery from $$C$$ to $$\phi(C)$$ which admits a $$\phi$$-periodic continuation to a gallery (3), giving rise to a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{r},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$, where $$r=\ell(\phi)$$. □ 5 Exceptional groups of type $$\tilde{E}_6$$ and $$\tilde{E}_7$$ Let $$G$$ be the group of $${\mathbb Q}_p$$-rational points of a $${\mathbb Q}_p$$-split connected reductive group over $${\mathbb Q}_p$$ with connected center $$Z$$. Fix a maximal $${\mathbb Q}_p$$-split torus $$T$$ and define $$\Phi$$, $$N(T)$$, $$W$$, $$\widehat{W}$$, and $$W_{{\rm aff}}$$ as before. 5.1 Affine root system $$\tilde{E}_6$$ Assume that the root system $$\Phi$$ is of type $$E_6$$. Following [1] (for the indexing) we then have generators $$s_1,\ldots,s_6$$ of $$W$$ and $$s_0,\ldots,s_6$$ of $$W_{{\rm aff}}$$ such that   (s1s3)3=(s3s4)3=(s4s5)3=(s5s6)3=(s2s4)3=(s0s2)3=1 and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u3=1 and us4u−1=s4,us3u−1=s5,us5u−1=s2,us2u−1=s3,us1u−1=s6,us6u−1=s0,us0u−1=s1. (33) (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the three-element subgroup $$W_{\Omega}=\{1,u,u^2\}$$.) Let $$e_1,\ldots,e_8$$ denote the standard basis of $${\mathbb R}^8$$. We use the standard inner product $$\langle.,.\rangle$$ on $${\mathbb R}^8$$ to view both the root system $$\Phi$$ as well as its dual $$\Phi^{\vee}$$ as living inside $${\mathbb R}^8$$. We choose a positive system $$\Phi^+$$ in $$\Phi$$ such that, as in [1], the simple roots are $$\alpha_1=\alpha_1^{\vee}=\frac{1}{2}(e_1+e_8-e_2-e_3-e_4-e_5-e_6-e_7)$$, $$\alpha_2=\alpha_2^{\vee}=e_2+e_1$$, $$\alpha_3=\alpha_3^{\vee}=e_2-e_1$$, $$\alpha_4=\alpha_4^{\vee}=e_3-e_2$$, $$\alpha_5=\alpha_5^{\vee}=e_4-e_3$$, $$\alpha_6=\alpha_6^{\vee}=e_5-e_4$$ while the negative of the highest root is $$\alpha_0=\alpha_0^{\vee}=\frac{1}{2}(e_6+e_7-e_1-e_2-e_3-e_4-e_5-e_8)$$. The set of positive roots is   Φ+={ej±ei|1≤i<j≤5}∪{12(−e6−e7+e8+∑i=15(−1)νiei)|∑i=15νi even}. We lift $$u$$ and the $$s_i$$ to elements $$\dot{u}$$ and $$\dot{s}_i$$ in $$N(T)$$. We then put   ϕ=s˙2s˙4s˙3s˙1u˙−1∈N(T). (34) We define $$\nabla$$ as in Section 3.1. Proposition 5.1. There is a $$\tau\in \nabla$$ such that the pair $$(\phi,\tau)$$ satisfies the hypotheses of Lemma 3.1. More precisely, $$\phi$$ is power multiplicative, and for the minuscule fundamental (co)weight $$\tau=\omega_1=\frac{2}{3}(e_8-e_7-e_6)\in \nabla$$ we have $$\phi^{12}=\tau^3$$ in $$W_{{\rm aff}}$$. □ Proof. (Here $$\tau^3$$ designates the three-fold iterate of translation by $$\tau=\omega_1$$, i.e., translation by the element $$3\omega_1$$ of the root lattice.) We define the subset   Φ(ω1)={12(−e6−e7+e8+∑i=15(−1)νiei)|∑i=15νi even} of $$\Phi^+$$. We compute $$\langle\beta,3\omega_1\rangle=3$$ for each $$\beta\in \Phi(\omega_1)$$, but $$\langle\beta,3\omega_1\rangle=0$$ for each $$\beta\in \Phi^+-\Phi(\omega_1)$$. This means that for each $$\beta\in \Phi(\omega_1)$$ the translation by $$3\omega_1$$ crosses $$3$$ walls parallel to the hyperplane corresponding to $$\beta$$, and that for each $$\beta\in \Phi^+-\Phi(\omega_1)$$ it crosses no wall parallel to the hyperplane corresponding to $$\beta$$. As $$\Phi(\omega_1)$$ contains 16 elements it follows that $$\ell(\tau^3)=\ell(3\omega_1)=3\cdot 16=48$$. On the other hand, in the Appendix we explain how one can prove $$\phi^{12}=3\omega_1$$ (as elements in $$W_{{\rm aff}}$$). Combining these two facts we deduce $$\ell(\phi^{12})=48$$ and hence that $$\phi$$ (which a priori has length $$\le 4$$) is power multiplicative.■ Remark. Instead of verifying $$\phi^{12}=3\omega_1$$ by means of a direct (computer) computation, one might try to prove that the affine transformation $$\phi^{12}$$ on $$A$$ crosses exactly the same walls as does $$3\omega_1$$ (which are described in the above proof). □ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{4},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Similarly, we may replace $$\phi$$ by its third power $$\phi^{3}$$ which (in contrast to $$\phi$$) is an element of $$W_{\rm aff}$$ (modulo $$T_0$$). It yields a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{12},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. As in our treatment of the cases $$C$$, $$B$$, $$D$$, and $$A$$, this functor identifies the set of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules bijectively with a set of certain $$E$$-symmetric étale $$(\varphi^{12},\Gamma)$$-modules over $${k}_{\mathcal E}$$ of dimension $$3$$. We leave the details to the reader. Remarks. (a) Dual to the above choice of $$\phi$$ is the choice   ϕ=s˙2s˙4s˙5s˙6u˙∈N(T). (35) For this choice, Proposition 5.1 holds true verbatim the same way, but now with the minuscule fundamental (co)weight $$\tau=\omega_6=\frac{1}{3}(3e_5+e_8-e_7-e_6)$$ with its corresponding subset (again containing $$16$$ elements)   Φ(ω6)={12(e5−e6−e7+e8+∑i=14(−1)νiei)|∑i=14νi even}∪{e5±ei|1≤i<5} of $$\Phi^+$$. Again see the Appendix. (b) For $$\phi$$ given by either (34) or (35), consider the corresponding reduced expression of $$\phi^3$$ in $$W_{{\rm aff}}$$ (obtained by three-fold concatenation and then conjugation of the $$s_i$$’s appearing with powers of $$u$$ so that no factor $$u$$ or $$u^2$$ remains). The number of occurrences of the $$s_i$$ are precisely the coefficients of the $$\alpha^{\vee}_i$$ in   α0∨+α1∨+α6∨+2α2∨+2α3∨+2α5∨+3α4∨=0. □ 5.2 Affine root system $$\tilde{E}_7$$ Assume that the root system $$\Phi$$ is of type $$E_7$$. Following [1] we then have generators $$s_1,\ldots,s_7$$ of $$W$$ and $$s_0,\ldots,s_7$$ of $$W_{{\rm aff}}$$ such that   (s0s1)3=(s1s3)3=(s3s4)3=(s4s5)3=(s5s6)3=(s6s7)3=(s2s4)3=1 and moreover $$(s_is_j)^2=1$$ for all other pairs $$i<j$$, and $$s_i^2=1$$ for all $$i$$. In the extended affine Weyl group $$\widehat{W}$$ we find (cf. [4]) an element $$u$$ of length $$0$$ with   u2=1 and us4u=s4,us2u=s2,us3u=s5,us6u=s1,us7u=s0,us5u=s3,us1u=s6,us0u=s7. (We have $$\widehat{W}=W_{{\rm aff}}\rtimes W_{\Omega}$$ with the two-element subgroup $$W_{\Omega}=\{1,u\}$$.) Let $$e_1,\ldots,e_8$$ denote the standard basis of $${\mathbb R}^8$$. We use the standard inner product $$\langle.,.\rangle$$ on $${\mathbb R}^8$$ to view both the root system $$\Phi$$ as well as its dual $$\Phi^{\vee}$$ as living inside $${\mathbb R}^8$$. We choose a positive system $$\Phi^+$$ in $$\Phi$$ such that, as in [1], the simple roots are $$\alpha_1=\alpha_1^{\vee}=\frac{1}{2}(e_1+e_8-e_2-e_3-e_4-e_5-e_6-e_7)$$, $$\alpha_2=\alpha_2^{\vee}=e_2+e_1$$, $$\alpha_3=\alpha_3^{\vee}=e_2-e_1$$, $$\alpha_4=\alpha_4^{\vee}=e_3-e_2$$, $$\alpha_5=\alpha_5^{\vee}=e_4-e_3$$, $$\alpha_6=\alpha_6^{\vee}=e_5-e_4$$, $$\alpha_7=\alpha_7^{\vee}=e_6-e_5$$ while the negative of the highest root is $$\alpha_0=\alpha_0^{\vee}=e_7-e_8$$. The set of positive roots is   Φ+={ej±ei|1≤i<j≤6}∪{e8−e7}∪{12(e8−e7+∑i=16(−1)νiei)|∑i=16νi odd}. We lift $$u$$ and the $$s_i$$ to elements $$\dot{u}$$ and $$\dot{s}_i$$ in $$N(T)$$. We then put   ϕ=s˙1s˙3s˙4s˙2s˙5s˙4s˙3s˙1s˙0u˙∈N(T). We define $$\nabla$$ as in Section 3.1. Proposition 5.2. There is a $$\tau\in \nabla$$ such that the pair $$(\phi,\tau)$$ satisfies the hypotheses of Lemma 3.1. More precisely, $$\phi$$ is power multiplicative, and for the minuscule fundamental (co)weight $$\tau=\omega_7=e_6+\frac{1}{2}(e_8-e_7)\in \nabla$$ we have $$\phi^{6}=\tau^2$$ in $$W_{{\rm aff}}$$. □ Proof. Exactly the same as for Proposition 5.1. The corresponding subset in $$\Phi^+$$ is   {12(e6+e8−e7+∑i=15(−1)νiei)|∑i=15νi odd}∪{e8−e7}∪{e6±ei|1≤i<6}. It contains 27 elements, thus $$\ell(2\omega_7)=54$$. For a computer proof of $$\phi^{6}=2\omega_7$$ see the Appendix.■ As explained in Section 3.2 we now obtain a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{9},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Similarly, we may replace $$\phi$$ by its square $$\phi^{2}$$ which (in contrast to $$\phi$$) is an element of $$W_{\rm aff}$$ (modulo $$T_0$$). It yields a functor from $${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$$ to the category of $$(\varphi^{18},\Gamma)$$-modules over $${\mathcal O}_{\mathcal E}$$. Again this functor identifies the set of standard supersingular $${\mathcal H}(G,I_0)_k$$-modules bijectively with a set of certain $$E$$-symmetric étale $$(\varphi^{18},\Gamma)$$-modules over $${k}_{\mathcal E}$$ of dimension $$2$$. We leave the details to the reader. Remark. The number of occurrences of the $$s_i$$ in $$\phi^2\in W_{{\rm aff}}$$ (cf. the case $$\tilde{E}_6$$) are the coefficients of the $$\alpha^{\vee}_i$$ in   α0∨+α7∨+2(α1∨+α2∨+α6∨)+3(α3∨+α5∨)+4α4∨=0. □ Appendix A.1 Verification of the statement $$\phi^{12}=3\omega_1$$ in the proof of Proposition 5.1 In the computer algebra system sage, the input R$$\,{=}\,$$RootSystem([“E”,6,1]).weight_lattice() Lambda$$\,{=}\,$$R.fundamental_weights() omega1$$\,{=}\,$$Lambda[1]-Lambda[0] R.reduced_word_of_translation(3*omega1) prompts the output   [0,2,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,0,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1]. (36) By definition of the function reduced_word_of_translation this means $$s^*_{i_1}\cdots s^*_{i_{48}}=3\omega_1$$, with the string $$[i_1,\ldots,i_{48}]$$ as given by (A.1). Here $$s_i^*=s_i$$ for $$1\le i\le 6$$, but $$s_0^*$$ denotes the affine reflection in the outer face of Bourbaki’s fundamental alcove $$A$$. Since we deviate from these conventions in that our $$s_0$$ is the affine reflection in the outer face of the negative$$C=-A$$ of $$A$$, we must modify the above string (A.1) as follows. First, writing $$s_i^{**}=s_0^*s_i^*s_0^*$$ for $$0\le i\le 6$$, conjugating the factors in the previous word by $$s_0^*$$ and commuting some of its factors where allowed, the above says $$s^{**}_{j_1}\cdots s^{**}_{j_{48}}=3\omega_1$$ where the string $$[j_1,\ldots,j_{48}]$$ is given by   [2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1,2,4,5,6,3,4,2,0,5,4,3,1]. (37) The $$s_i^{**}$$ are precisely the reflections in the codimension $$1$$ faces of $$s_0^*A$$. But $$s_0^*A$$ is a translate of $$C$$, and under this translation, the reflection $$s_0^{**}=s_0^*$$ corresponds to $$s_0$$, whereas for $$1\le i\le 6$$ the reflection $$s_i^{**}$$ corresponds to $$w_0s_iw_0$$, where $$w_0$$ is the longest element of $$W$$. We have $$w_0s_iw_0=s_i$$ for $$i\in\{0,2,4\}$$, but $$w_0s_3w_0=s_5$$ and $$w_0s_1w_0=s_6$$. Thus, we obtain $$s_{k_1}\cdots s_{k_{48}}=3\omega_1$$ where the string $$[k_1,\ldots,k_{48}]$$ is obtained from the string (A.2) by keeping its entry values $$0, 2$$, and $$4$$, while exchanging the entry values $$3$$ with $$5$$ and $$1$$ with $$6$$. Using formulae (33) one checks that $$s_{k_1}\cdots s_{k_{48}}=\phi^{12}$$. $$\Box$$ A.2 Verification of the statement $$\boldsymbol{\phi^{12}=3\omega_6}$$ for $$\boldsymbol{\phi}$$ given by (35) The argument is the same as in Proposition 5.1. The string returned by sage to R$$\,{=}\,$$RootSystem([“E”,6,1]).weight_lattice(), Lambda$$\,{=}\,$$R.fundamental_weights(), omega6$$\,{=}\,$$Lambda[6]-Lambda[0], R.reduced_word_of_translation(3*omega6) reads   [0,2,4,3,1,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,0,3,4,2,5,4,3,1,6,5,4,2,3,4,5,6]. A.3 Verification of the statement $$\boldsymbol{\phi^{6}=2\omega_7}$$ in the proof of Proposition 5.2 The argument is the same as in Proposition 5.1. The string returned by sage to R$$\,{=}\,$$RootSystem([“E”,7,1]).weight_lattice(), Lambda$$\,{=}\,$$R.fundamental_weights(), omega7$$\,{=}\,$$Lambda[7]-Lambda[0], R.reduced_word_of_translation(2*omega7) reads   [0,1,3,4,2,5,4,3,1,0,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,0,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,5,4,2,3,4,5,6,7]. (38) Acknowledgments I would like to thank Laurent Berger for a helpful discussion related to this work. I am very grateful to the referees for the careful reading of the manuscript and the detailed suggestions for improvements. Footnotes 1 (Pick $$g\in N(T)$$ acting as the reflection in the affine hyperplane which contains the one-codimensional facet shared by $$D_1$$ and $$D_2$$. Then $$g D_1=D_2$$ and hence $$I_{D_2}=g I_{D_1}g^{-1}$$, so we can read off all these claims.) References [1] Bourbaki N. Groupes at Algèbres de Lie, ch. 4, 5, 6 / ch. 7, 8, 9 . Springer-Verlag, Berlin Heidelberg, 2006. [2] Grosse-Klönne E. “From pro-$$p$$-Iwahori Hecke modules to $$(\Phi,\Gamma)$$-modules, I.” Duke Mathematical Journal  165, no. 8 ( 2016): 1529– 95. [3] He X. and Nie. S. “Minimal length elements of extended affine Weyl groups.” Compositio Mathematica  150, no. 11 ( 2014): 1903– 27. Google Scholar CrossRef Search ADS   [4] Iwahori N. and Matsumoto. H. “On some Bruhat decomposition and the structure of the Hecke rings of $$p$$-adic Chevalley groups.” Publications Mathèmatiques de lĺHèS  25 ( 1965): 5– 48. Google Scholar CrossRef Search ADS   [5] Jantzen J. C. Representations of Algebraic Groups . Mathematical Surveys and Monographs 107, 2nd ed. American Mathematical Society, USA, 2003. [6] Koziol K. “Pro-$$p$$-Iwahori invariants for $${\rm SL}_2$$ and $$L$$-packets of Hecke modules.” International Mathematics Research Notices  2016, no. 4 ( 2016): 1090- 125. Google Scholar CrossRef Search ADS   [7] Ollivier R. “Compatibility between Satake and Bernstein-type isomorphisms in characteristic $$p$$.” Algebra and Number Theory  5 ( 2014): 1071– 111. Google Scholar CrossRef Search ADS   [8] Vignèras M. F. “Pro-$$p$$-Iwahori Hecke ring and supersingular $$\overline{\mathbb F}_p$$-representations.” Mathematische Annalen  331, no. 3 ( 2005): 523- -56 and Mathematische Annalen 333, no. 3 ( 2005): 699– 701. Google Scholar CrossRef Search ADS   [9] Vignèras M. F. “The pro-$$p$$-Iwahori Hecke algebra of a reductive $$p$$-adic group III.” Journal of the Institute of Mathematics of Jussieu  (to appear). © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

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Published: Feb 1, 2018

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