# On the Acceptable Elements

On the Acceptable Elements Abstract In this article, we study the set $$B(G, \{\mu\})$$ of acceptable elements for any $$p$$-adic group $$G$$. We show that $$B(G, \{\mu\})$$ contains a unique maximal element and the maximal element is represented by an element in the admissible subset of the associated Iwahori–Weyl group. 1 Introduction Let $$F$$ be a finite field extension of $$\mathbb Q_p$$ and $$L$$ be the completion of the maximal unramified extension of $$F$$. Let $$G$$ be a connected reductive algebraic group over $$F$$ and $$\sigma$$ be the Frobenius morphism. We denote by $$B(G)$$ the set of $$\sigma$$-conjugacy classes of $$G(L)$$. The set $$B(G)$$ is classified by Kottwitz in [11] and [12]. This classification generalizes the Dieudonnè–Manin classification of isocrystals by their Newton polygons. Let $$\tilde W$$ be the Iwahori–Weyl group of $$G$$ over $$L$$. Let $$\{\mu\}$$ be a geometric conjugacy class of cocharacters of $$G$$. Let $$\text{Adm}(\{\mu\}) \subseteq \tilde W$$ be the admissible subset of $$\tilde W$$ ([13] and [16]) and $$B(G, \{\mu\})$$ be the finite subset of $$B(G)$$ defined by the group-theoretic version of Mazur’s theorem [12, Section 6]. The main result of this article is as follows. Theorem 1.1. The set $$B(G, \{\mu\})$$ contains a unique maximal element and this element is represented by an element in $$\text{Adm}(\{\mu\})$$. □ For quasi-split groups, this is obvious as the unique maximal element of $$B(G, \{\mu\})$$ is represented by a translation element. However, it is much more complicated for non quasi-split groups. This result is an important ingredient in the proof [8] of the Kottwitz–Rapoport conjecture [14, Conjecture 3.1] and [16, Conjecture 5.2] on the union of affine Deligne–Lusztig varieties. The knowledge of the explicit description of the maximal element of $$B(G, \{\mu\})$$ is also useful in the study of the $$\mu$$-ordinary locus, the most general Newton stratum, of Shimura varieties. In fact, Theorem 1.1 is a statement on the Iwahori–Weyl group $$\tilde W$$, the automorphism on $$\tilde W$$ induced from the Frobenius morphism $$\sigma$$ of $$G$$ and a cocharacter $$\mu$$ in $$\{\mu\}$$. In Section 2, we introduce a set $$B(\tilde W, \mu, \sigma)$$ (for any diagram automorphism $$\sigma$$ on $$\tilde W$$) and reformulate Theorem 1.1 as a statement on the triple $$(\tilde W, \mu, \sigma)$$. The relation between $$B(G, \{\mu\})$$ and $$B(\tilde W, \mu, \sigma)$$ is discussed in the Appendix. This reformulation allows us to relate different diagram automorphisms of the Iwahori–Weyl groups, which plays an essential role in the proof. We show that the set $$B(\tilde W, \mu, \sigma)$$ contains a unique maximal element in Section 3. It requires more work to show that this maximal element is represented by an element in the admissible set. We show in Section 4 that it suffices to consider the superbasic case and in Section 5 that it suffices to consider the irreducible case. In Section 6, we prove the statement for the irreducible superbasic case (that is, $$\sigma$$ is a diagram automorphism of order $$n$$ for the Iwahori–Weyl group of $$PGL_n$$). This completes the proof of Theorem 1.1. 2 Preliminaries 2.1 Let $$\mathfrak R=(X^*, R, X_*, R^\vee, \Pi)$$ be a based reduced root datum, where $$R \subseteq X^*$$ is the set of roots, $$R^\vee \subseteq X_*$$ is the set of coroots and $$\Pi \subseteq R$$ is the set of simple roots. Let $$\langle ~ , ~ \rangle : X^* \times X_* \rightarrow \mathbb Z$$ be the natural perfect pairing between $$X^*$$ and $$X_*$$. Let $$V=X_* \otimes \mathbb R$$. For any $$\alpha \in R$$, we have a reflection $$s_\alpha$$ on $$V$$ sending $$v$$ to $$v-\langle \alpha, v\rangle \alpha^\vee$$. The reflections $$s_\alpha$$ generate the finite Weyl group$$W_0$$ of $$R$$. Let $$\mathbb S=\{s_\alpha; \alpha \in \Pi\}$$ be the set of simple reflections. Then $$(W_0, \mathbb S)$$ is a Coxeter system. For any $$J \subseteq \mathbb S$$, let $$W_J$$ be the subgroup of $$W_0$$ generated by $$J$$ and $${}^J W_0=\{w \in W_0;$$$$w=\min(W_J w)\}$$. The closed dominant chamber is the set   C={v∈V;⟨α,v⟩≥0 for every α∈Π}. Then for any $$v \in V$$, the set $$\{w(v); w \in W_0\}$$ contains a unique element in $$\mathfrak C$$. We denote this element by $$\bar v$$. 2.2 Set $$\tilde R=R \times \mathbb Z$$. For $$(\alpha, k) \in \tilde R$$, we have an affine root $$\tilde \alpha=\alpha+k$$ and an affine reflection $$s_{\tilde \alpha}$$ on $$V$$ sending $$v$$ to $$v-(\langle\alpha, v\rangle - k) \alpha^\vee$$. For any affine root $$\tilde \alpha$$, let $$H_{\tilde \alpha}$$ be the hyperplane in $$V$$ fixed by the reflection $$s_{\tilde \alpha}$$. Set   Wa=ZR∨⋊W0={tλw;λ∈ZR∨,w∈W0},W~=X∗⋊W0={tλw;λ∈X∗,w∈W0}. We call $$W_a$$ the affine Weyl group and $$\tilde W$$ the extended affine Weyl group. Let $$\text{Aff}(V)$$ be the group of affine transformations on $$V$$. We realize both $$W_a$$ and $$\tilde W$$ as subgroups of $$\text{Aff}(V)$$, where $$t^\lambda$$ acts by translation $$v \mapsto v+\lambda$$ on $$V$$. We may identify $$W_a$$ with the subgroup of $$\text{Aff}(V)$$ generated by the affine reflections. Let $$R^+ \subseteq R$$ be the set of positive roots determined by $$\Pi$$. The base alcove is the set   a={v∈V;0<⟨α,v⟩<1 for every α∈R+}. The set of positive affine roots is $$\{\alpha+k; \alpha \in R^+, k \geq 1\} \cup \{-\alpha+k; \alpha \in R^+, k \geq 0\}$$. The affine simple roots are $$-\alpha$$ for $$\alpha \in \Pi$$ and $$\beta+1$$, where $$\beta$$ runs over maximal positive roots in $$R$$. Note that the positive roots in $$R$$ are not positive as affine roots. The hyperplanes $$H_{\tilde \alpha}$$, for the affine simple roots $$\tilde \alpha$$, are exactly the walls of the base alcove $$\mathbf a$$. Let $$\widetilde{\mathbb{S}}$$ be the set of $${{s}_{{\tilde{\alpha }}}}$$, where $$\tilde \alpha$$ runs over affine simple roots. Then $$\mathbb{S}\subseteq \widetilde{\mathbb{S}}$$ and $$({{W}_{a}},\widetilde{\mathbb{S}})$$ is a Coxeter group. Let $$\Omega$$ be the isotropy group in $$\tilde W$$ of the base alcove $$\mathbf a$$. Then $$\tilde W=W_a \rtimes \Omega$$. We extend the Bruhat order on $$W_a$$ to $$\tilde W$$ as follows: for $$w, w' \in W_a$$ and $$\tau, \tau' \in \Omega$$, we say that $$w \tau \le w' \tau'$$ if $$\tau=\tau'$$ and $$w \le w'$$ (with respect to the Bruhat order on the Coxeter group $$W_a$$). We put $$\ell(w \tau)=\ell(w)$$. 2.3 Let $$\sigma \in \text{Aff}(V)$$ be an automorphism of finite order such that $$\sigma(\mathbf a)=\mathbf a$$ and the conjugation action of $$\sigma$$ stabilizes $$\tilde W$$. Then $$\sigma$$ induces a bijection on the set of walls of $$\mathbf{a}$$ and hence a bijection on $$\widetilde{\mathbb{S}}$$. Let $$\varsigma \in GL(V)$$ denote the linear part of $$\sigma$$ with respect to the decomposition $$\text{Aff}(V)=V \rtimes GL(V)$$. Then $$\sigma$$ acts by conjugation on the translation subgroup of $$\text{Aff}(V)$$ via $$\varsigma$$:   Ad(σ)(tξ)=tς(ξ) for ξ∈V. Since we assume that $$\sigma$$ normalizes $$\tilde W$$, it follows that $$\varsigma$$ stabilizes $$X_*$$ and normalizes $$W_0$$. The $$\sigma$$-conjugation action on $$\tilde W$$ is defined by $$w \cdot_\sigma w'=w w' \text{Ad}(\sigma)(w)^{-1}$$. We have two invariants on the $$\sigma$$-conjugacy classes. Since $$\sigma(\mathbf a)=\mathbf a$$, the conjugation action of $$\sigma$$ stabilizes $$\Omega$$. Let $$\Omega_\sigma$$ be the set of $$\sigma$$-coinvariants on $$\Omega$$. The Kottwitz map $$\kappa_{\tilde W, \sigma}: \tilde W \to \Omega_\sigma$$ is obtained by composing the natural projection map $$\tilde W \to \tilde W/W_a \cong \Omega$$ with the projection map $$\Omega \to \Omega_\sigma$$. It is constant on each $$\sigma$$-conjugacy class of $$\tilde W$$. This gives one invariant. Another invariant is given by the Newton map. For any $$w \in \tilde W$$, we consider the element $$w \sigma \in \text{Aff}(V)$$. There exists $$n \in \mathbb N$$ such that $$(w \sigma)^n=t^\xi$$ for some $$\xi \in X_*$$. Let $$\nu_{w, \sigma}=\xi/n$$ and $$\bar \nu_{w, \sigma}$$ be the unique dominant element in the $$W_0$$-orbit of $$\nu_{w, \sigma}$$. We call $$\bar \nu_{w, \sigma}$$ the Newton point of $$w$$ (with respect to the $$\sigma$$-conjugation action). It is known that $$\nu_{w, \sigma}$$ is independent of the choice of $$n$$ and $$\bar \nu_{w, \sigma}=\bar \nu_{w', \sigma}$$ if $$w$$ and $$w'$$ are $$\sigma$$-conjugate. Moreover, $$w \sigma t^\xi=t^\xi w \sigma \in \text{Aff}(V)$$. Hence $$w \text{Ad}(\sigma)(t^{\xi})=t^\xi w$$. Therefore $$\varsigma(\xi)$$ and $$\xi$$ are in the same $$W_0$$-orbit and   ν¯w,σ=ς(νw,σ)¯. (a) 2.4 Let $$\mu$$ be a dominant cocharacter, that is $$\mu \in X_* \cap \mathfrak C$$. The $$\mu$$-admissible set is defined as   Adm(μ)={w∈W~;w≤tx(μ) for some x∈W0}. The partial order on $$\mathfrak C$$ is defined as follows. Let $$v, v' \in \mathfrak C$$. We say that $$v \leq v'$$ if $$v'-v \in \sum_{\alpha \in \Pi} \mathbb R_{\geq 0} \alpha^\vee$$. Let $$N$$ be the order of $$\sigma$$. For $$\mu \in X_*$$, we define   μσ♢=1N∑i=0N−1ςi(μ)¯∈Cμσ♣=νtμ,σ=1N∑i=0N−1ςi(μ). If $$\sigma(0)=0$$, then $$\mu^\diamondsuit_\sigma=\mu^\clubsuit_\sigma$$. Set   B(W~,μ,σ)={ν¯w,σ;w∈W~,κW~,σ(w)=κW~,σ(tμ),ν¯w,σ≤μσ♢}. The elements in $$B(\tilde W, \mu, \sigma)$$ are called the acceptable elements for $$\mu$$. The main result of this article is as follows. Theorem 2.1. (1) The set $$B(\tilde W, \mu, \sigma)$$ contains a unique maximal element $$\nu$$ (with respect to the partial order $$\leq$$ on $$\mathfrak C$$). (2) There exists an element $$w \in \text{Adm}(\mu)$$ with $$\bar \nu_{w, \sigma}=\nu$$. □ The relation between the sets $$B(\tilde W, \mu, \sigma)$$ and $$B(G, \{\mu\})$$ will be discussed in the Appendix. 3 The Maximal Element in $$B(\tilde W, \mu, \sigma)$$ 3.1 Let $$\mathfrak R_{ad}=(\mathbb Z R, R, X_*^{ad}, R^\vee, \Pi)$$ be the root datum of the adjoint group of the reductive group with root datum $$\mathfrak R$$. Here $$X_*^{ad}$$ is the dual lattice of $$\mathbb Z R$$. The perfect pairing $$\langle ~ , ~ \rangle : X^* \times X_* \rightarrow \mathbb Z$$ induces a natural map $$\pi: X_* \otimes \mathbb R \to X_*^{ad} \otimes \mathbb R$$. Set $$\tilde W_{ad}=X_*^{ad} \rtimes W_0$$. Then the map $$\pi$$ induces a natural map $$\pi: \tilde W \to \tilde W_{ad}$$. Lemma 3.1. Let $$v \in X_*^{ad} \otimes \mathbb R$$ and $$\hat{v}$$, $$\hat{v}'$$ be two lifts of $$v$$ under $$\pi$$. Then $$\pi \circ \sigma(\hat{v})=\pi \circ \sigma(\hat{v}')$$. □ Proof Note that $$\sigma(\hat{v})=\sigma(\hat{v}')+\sigma(\hat{v}-\hat{v}')-\sigma(0)$$. Since $$\varsigma$$ normalizes $$W_0$$, it gives a permutation of the hyperplanes $$H_\alpha=\{v \in V; \langle\alpha, v\rangle=0\}$$ for $$\alpha \in R$$. As $$\hat{v}-\hat{v}'$$ lies in the intersection of these hyperplanes, we have $$\sigma(\hat{v}-\hat{v}')-\sigma(0)=\varsigma(\hat{v}-\hat{v}')$$ is still in this intersection. In other words, $$\pi(\sigma(\hat{v}-\hat{v}')-\sigma(0))=0$$. Thus $$\pi \circ \sigma(\hat{v})=\pi \circ \sigma(\hat{v}')$$. ■ Now we define a map $$\sigma_{ad}: X_*^{ad} \otimes \mathbb R \to X_*^{ad} \otimes \mathbb R$$ by $$v \mapsto \pi \circ \sigma(\hat{v})$$, where $$\hat{v} \in V$$ is a lift of $$v \in X_*^{ad} \otimes \mathbb R$$ under $$\pi$$. By Lemma 3.1, $$\sigma_{ad}$$ is well defined. The affine transformation $$\sigma_{ad}$$ on $$X_*^{ad} \otimes \mathbb R$$ induces a conjugation action on $$\tilde W_{ad}$$. It is easy to see that $$\pi(\nu_{w, \sigma})=\nu_{\pi(w), \sigma_{ad}}$$ for $$w \in \tilde W$$ and $$\pi$$ induces a bijection of posets from $$B(\tilde W, \mu, \sigma)$$ to $$B(\tilde W_{ad}, \pi(\mu), \sigma_{ad})$$. The map $$\pi$$ also induces a bijection of posets from $$\text{Adm}(\mu)$$ to $$\text{Adm}(\pi(\mu))$$. Thus Theorem 2.1 holds for $$B(\tilde W, \mu, \sigma)$$ if and only if it holds for $$B(\tilde W_{ad}, \pi(\mu), \sigma_{ad})$$. Lemma 3.2. If $$\mathfrak R=\mathfrak R_{ad}$$, then $$\Omega$$ acts simply transitively on the set of special vertices of $$\mathbf a$$. □ Remark 3.3. This Lemma is known to experts. We include a proof for the reader’s convenience. □ Proof First, the action of $$\Omega$$ on $$V$$ stabilizes the set of special vertices of $$\mathbf a$$. Let $$v \in V$$ be a special vertex of $$\mathbf a$$. By [1, VI Section 2 Prop. 3], $$v \in X_*$$. Since $$W_0$$ acts simply transitively on the set of chambers, there exists a unique $$x \in W_0$$ such that $$x t^{-v}$$ sends $$\mathbf a$$ to an alcove $$\mathbf a'$$ in the dominant chamber. Since $$x t^{-v}(v)=0$$, we deduce that $$0$$ lies in the closure of $$\mathbf a'$$. Note that $$\mathbf a$$ is the unique alcove in the dominant chamber whose closure contains $$0$$. Hence $$\mathbf a'=\mathbf a$$ and $$x t^{-v} \in \Omega$$. In other words, there exists an element in $$\Omega$$ sending $$v$$ to $$0$$. So $$\Omega$$ acts transitively on the set of special vertices of $$\mathbf a$$. Let $$w \in \tilde W$$. If $$w$$ preserves $$0$$, then $$w \in W_0$$. Note that $$\Omega \cap W_0=\{1\}$$. Thus the action of $$\Omega$$ on the set of special vertices is simply transitive. ■ 3.2 In the rest of this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$. By Lemma 3.2, there exists $$\tau \in \Omega$$ such that $$\sigma_0:=\tau^{-1} \sigma$$ preserves $$0 \in V$$. In the rest of the article, unless otherwise stated, we denote by $$\lambda$$ the dominant cocharacter with $$\tau \in t^\lambda W_0$$. Notice that $$\sigma_0$$ is a linear action on $$V$$ and the conjugation action of $$\sigma_0$$ stabilizes the subset $$\mathbb S$$ of $$\tilde{\mathbb S}$$. For simplicity, in the rest of the article, we will say $$\sigma_0$$-orbits in $$\mathbb S$$ instead of $$\text{Ad}(\sigma_0)$$-orbits in $$\mathbb S$$. %Then $$\sigma_0$$ stabilizes $$\mathbb S$$ inside $$\tilde{\mathbb S}$$ and is a diagram automorphism of $$W_0$$. In particular, $$\sigma_0$$ is a linear action on $$V$$. By definition, $$\nu_{w, \sigma}=\nu_{w \tau, \sigma_0}$$ for all $$w \in \tilde W$$. By 2.3 (a),   ν¯w,σ=ν¯wτ,σ0∈Cσ0:={v∈C;σ0(v)=v}. Lemma 3.4. If $$\xi \in X_* \cap \mathfrak C$$, then $$\xi^\diamondsuit_\sigma=\xi^\clubsuit_{\sigma_0}=\nu_{t^\xi, \sigma_0}$$. □ Proof Note that $$\sigma_0=\tau^{-1} \sigma$$ and is linear. We have $$\sigma_0=x \varsigma$$ for some $$x \in W_0$$. Since $$\varsigma$$ normalizes $$W_0$$, we have $$\sigma_0^i \in W_0 \xi^i$$ for any $$i \in \mathbb Z$$. Therefore $$\overline{\varsigma^i(\zeta)}=\overline{\sigma_0^i(\zeta)}$$. Since $$\sigma_0$$ stabilizes $$\mathfrak C$$, $$\xi^\diamondsuit_\sigma=\xi^\diamondsuit_{\sigma_0}=\xi^\clubsuit_{\sigma_0}$$. ■ 3.3 For any $$i \in \mathbb S$$, let $$\omega^\vee_i \in V$$ be the corresponding fundamental coweight and $$\alpha_i^\vee \in V$$ be the corresponding simple coroot. We denote by $$\omega_i, \alpha_i \in V^*$$ the corresponding fundamental weight and corresponding simple root, respectively. For each $$\sigma_0$$-orbit $$c$$ of $$\mathbb S$$, we set   ωc=∑i∈cωi. For any $$v \in \mathfrak C$$, we set   J(v)={s∈S;s(v)=v},I(v)=S∖J(v). If $$v=\sigma_0(v)$$, then both $$J(v)$$ and $$I(v)$$ are $$\sigma_0$$-stable. The following lemma is essentially contained in [2, Section 7.1]. Due to its importance, we provide a proof for completeness. Lemma 3.5. Let $$v \in \mathfrak C^{\sigma_0}$$. Then $$v =\nu_{w, \sigma}$$ for some $$w \in t^\mu W_a$$ if and only if $$\langle\omega_c, \mu^\clubsuit_{\sigma_0} + \lambda^\clubsuit_{\sigma_0}-v\rangle \in\mathbb Z$$ for any $$\sigma_0$$-orbit $$c$$ of $$I(v)$$. □ Proof Suppose $$\nu_{w, \sigma}=v$$. We have $$w \tau=t^\gamma x$$ for some $$\gamma \in X_*$$ and $$x \in W_0$$. By definition, $$(w\sigma)^n=t^{n v}$$ for some $$n \in \mathbb N$$. Since $$t^{n v}$$ commutes with $$w\sigma=t^\gamma x\sigma_0$$, we have $$x\sigma_0(v)=x(v)=v$$. Thus $$x \in W_{J(v)}$$. Let $$N_0$$ be the order of the finite subgroup of $$\text{Aff}(V)$$ generated by $$W_0$$ and $$\sigma_0$$. Then   νw,σ=νwτ,σ0=1N0∑k=0N0−1(xσ0)k(γ)=1N0∑k=0N0−1(xAd(σ0)(x)⋯Ad(σ0)k−1(x))σ0k(γ)∈1N0∑k=0N0−1σ0k(γ)+∑j∈J(v)Qαj∨=γσ0♣+∑j∈J(v)Qαj∨. If $$w \in t^\mu W_a$$, then $$w \tau \in t^{\mu+\lambda} W_a$$ and $$\mu+\lambda-\gamma \in \mathbb Z R^\vee$$. Hence   ⟨ωc,μσ0♣+λσ0♣−v⟩=⟨ωc,μσ0♣+λσ0♣−γσ0♣⟩=⟨ωc,μ+λ−γ⟩∈Z. On the other hand, suppose $$a_c=\langle\omega_c, \mu^\clubsuit_{\sigma_0} + \lambda^\clubsuit_{\sigma_0}-v\rangle \in \mathbb Z$$ for each $$\sigma_0$$-orbit $$c$$ of $$I(v)$$. We also set $$a_c=0$$ if $$c \nsubseteq I(v)$$. We construct an element $$w \in t^\mu W_a$$ such that $$\nu_{w,\sigma}=v$$. For each $$\sigma_0$$-orbit of $$J(v)$$, we choose a representative. Let $$y$$ be the product of these representatives (in some order). Then $$y$$ is a $$\sigma_0$$-twisted Coxeter element of $$W_{J(v)}$$ in the sense of [18, 7.3]. For each $$\sigma_0$$-orbit $$c$$ of $$I(v)$$, we choose a representative $$i_c$$. Let $$\alpha^\vee_{i_c}$$ be the corresponding simple coroot. Set $$\beta=\mu+\lambda-\sum_c a_c \alpha^\vee_{i_c}$$ and $$w=t^{\beta} y \tau^{-1} \in t^\mu W_a$$. Write $$\beta=h+r$$ with $$r \in \sum_{j \in J(v)} \mathbb Q \alpha_j^\vee$$ and $$h \in \sum_{i \in I(v)} \mathbb Q \omega_i^\vee$$. Then   νw,σ=νwτ,σ0=1N0∑k=0N0−1(yσ0)k(β)=hσ0♣+1N0∑k=0N0−1(yσ0)k(r)=hσ0♣=μσ0♣+λσ0♣−∑cac(αic∨)σ0♣−rσ0♣, where the fourth equality follows from [18, Lemma 7.4]. Hence for any $$\sigma_0$$-orbit $$c$$ of $$I(v)$$ and any $$j \in J(v)$$, we have   ⟨ωc,μσ0♣+λσ0♣−νw,σ⟩=⟨ωc,∑c′ac′(αic′∨)σ0♣⟩=ac and   ⟨αj,μσ0♣+λσ0♣−νw,σ⟩=⟨αj,μσ0♣+λσ0♣⟩=⟨αj,μσ0♣+λσ0♣−v⟩, which means $$\nu_{w,\sigma}=v$$ as desired. ■ Corollary 3.6. $$\mu^\diamondsuit_{\sigma}=\mu^\clubsuit_{\sigma_0} \in B(\tilde W, \mu, \sigma)$$ if and only if $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$ for any $$\sigma_0$$-orbit $$c$$ of $$I(\mu^\diamondsuit_{\sigma})$$. In this case, $$\mu^\diamondsuit_{\sigma}$$ is a priori the unique maximal element of $$B(\tilde W, \mu, \sigma)$$. □ 3.4 We follow [2, Section 6]. For any $$\sigma_0$$-stable subset $$B$$ of $$\mathfrak C$$, we define   C≥B={v∈Cσ0;v≥b,∀b∈B}. We say $$B$$ is reduced if $$C_{\geq B} \subsetneq C_{\geq B'}$$ for any $$\sigma_0$$-stable proper subset $$B' \subsetneq B$$. For any $$i \in \mathbb S$$, let   pr(i):V=Rωi∨⊕∑j≠iRαj∨→Rωi∨ be the projection map. Now we prove part (1) of Theorem 2.1. 3.5 Proof of Theorem 2.1 (1) By Section 3.1, it suffices to consider the case where $$\mathfrak R=\mathfrak R_{ad}$$. For any $$i \in \mathbb S$$, let $$c$$ denote the $$\sigma_0$$-orbit of $$i$$, and define $$e_i \in \mathbb Q \omega_i^\vee$$ by   ⟨ωi,ei⟩=1#cmax({t∈⟨ωc,μσ0♣+λσ0♣⟩+Z;t≤⟨ωc,μσ0♣⟩}∪{0}). Let $$E_0=\{e_i; i \in \mathbb S\}$$. It is easy to prove by induction on the number of $$\sigma_0$$-orbits on $$E_0$$ that there exists a $$\sigma_0$$-stable subset $$E$$ of $$E_0$$ which is reduced and satisfies $$C_{\geq E}=C_{\geq E_0}$$. Let $$I(E)=\{i \in \mathbb S; e_i \in E\}$$. By [2, Theorem 6.5] In fact, we use here a “$$\sigma_0$$-fixed” version of [2, Theorem 6.5], which can be proved in the same way as in loc.cit., there exists an element $$\nu \in C_{\geq E}$$ defined by $$I(\nu)=I(E)$$ and $$\langle\omega_j, \nu\rangle=\langle\omega_j, e_j\rangle$$ for $$j \in I(E)$$, which satisfies $$C_{\geq \nu}=C_{\geq E}=C_{\geq E_0}$$. Since $$\mu^\diamondsuit_\sigma=\mu^\clubsuit_{\sigma_0} \in C_{\geq E_0}$$, we have $$\nu \leq \mu^\diamondsuit_\sigma$$. By Lemma 3.5, $$\nu \in B(\tilde W, \mu, \sigma)$$. (In fact, we use here a “$$\sigma_0$$-fixed” version of [2, Theorem 6.5], which can be proved in the same way as in loc.cit.) Since $$\nu \in C_{\geq E_0}$$, $$\nu \geq e_i$$ for any $$i \in \mathbb S$$. Therefore, for any $$\sigma_0$$-orbit $$c$$ of $$\mathbb S$$, we have   ⟨ωc,μσ0♣⟩≥⟨ωc,ν⟩≥∑j∈c⟨ωj,ej⟩≥⟨ωc,μσ0♣+λσ0♣⟩−⌈⟨ωc,λσ0♣⟩⌉, (a) where the last inequality follows from our definition of $$e_j$$ for $$j \in \mathbb S$$. Let $$\nu' \in B(\tilde W,\mu, \sigma)$$. Set $$E(\nu')=\{pr_{(j)}(\nu'); j \in I(\nu')\}$$. By Lemma 3.5 and the inequality $$\nu' \leq \mu^\diamondsuit_\sigma=\mu^\clubsuit_{\sigma_0}$$, we have, for any $$\sigma_0$$-orbit $$c$$ of $$I(\nu')$$ and $$j \in c$$, that   #c⋅⟨ωj,pr(j)(ν′)⟩=#c⋅⟨ωj,ν′⟩=⟨ωc,ν′⟩∈⟨ωc,μσ0♣+λσ0♣⟩+Z and   #c⋅⟨ωj,pr(j)(ν′)⟩≤#c⋅⟨ωj,μσ0♣⟩=⟨ωc,μσ0♣⟩. So $$\langle\omega_j, pr_{(j)}(\nu')\rangle \leq \langle\omega_j, e_j\rangle$$, that is, $$pr_{(j)}(\nu') \leq e_j \leq \nu$$ for $$j \in I(\nu')$$. By [2, Lemma 6.2 (i)], we deduce that $$\nu' \leq \nu$$. Therefore $$\nu$$ is the unique maximal element of $$B(\tilde W,\mu, \sigma)$$. 4 Reduction to the Superbasic Case 4.1 In the rest of the article, we prove Theorem 2.1 (2), beginning in this section with a reduction step to the superbasic case. For any element $$w \sigma^i$$ with $$w \in \tilde W$$ and $$i \in \mathbb Z$$, we put $$\ell(w \sigma^i)=\ell(w)$$. This is well-defined since $$\sigma(\mathbf a)=\mathbf a$$. Let $$\epsilon=w \sigma^i$$ with $$\ell(\epsilon)=0$$. Then the conjugation action of $$\epsilon$$ on $$\tilde W$$ sends simple reflections to simple reflections. We say that $$\epsilon$$ is superbasic (for $$\tilde W$$) if each $$\text{Ad}(\epsilon)$$-orbit on $$\tilde{\mathbb S}$$ is a union of connected components of the affine Dynkin diagram of $$\tilde W$$. By [9, 3.5], $$\epsilon$$ is superbasic if and only if $$W_a=W_1^{m_1} \times \cdots \times W_l^{m_l}$$, where each $$W_i$$ is an extended affine Weyl group of type $$\tilde A_{n_i-1}$$ and $$\epsilon$$ gives an order $$n_i m_i$$ permutation on the set of simple reflections of $$W_i^{m_i}$$. 4.2 Let $$J \subseteq \mathbb S$$. Let $$\tilde W_J=X_* \rtimes W_J$$ be the corresponding parabolic subgroup of $$\tilde W$$. This is the extended affine Weyl group associated to the root datum $$\mathfrak R_J=(X^*, R_J, X_*, R^\vee_J, \Pi_J)$$, where $$\Pi_J$$ is the subset of simple roots corresponding to $$J$$ and $$R_J \subseteq R$$ is the set of roots spanned by $$\Pi_J$$. Set $$R^+_J=R_J \cap R^+$$. Let $$\mathbf a_J=\{v \in V; 0 < \langle \alpha, v\rangle < 1 \text{ for every } \alpha \in R^+_J\}$$ be the base alcove associated to $$\tilde W_J$$. The set of positive affine roots $$\tilde R_J$$ for $$\tilde W_J$$ is $$\{\alpha+k; \alpha \in R^+_J, k \geq 1\} \cup \{-\alpha+k; \alpha \in R^+_J, k \geq 0\}$$. The affine simple roots for $$\tilde W_J$$ are $$-\alpha$$ for $$\alpha \in \Pi_J$$ and $$\beta+1$$, where $$\beta$$ runs over maximal positive roots in $$R_J^+$$. Note that the positive roots in $$R_J$$ are not positive as affine roots in $$\tilde R_J$$. We denote by $$\le_J$$ and $$\ell_J$$ the Bruhat order and length function on $$\tilde W_J$$. Although $$\tilde W_J$$ is a subgroup of $$\tilde W$$, $$\le_J$$ and $$\ell_J$$ can be quite different from the restrictions of $$\le$$ and $$\ell$$ to $$\tilde W_J$$. 4.3 In the rest of this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$. We take $$\tilde t \in \Omega$$ and $$\sigma_0=\tau^{-1} \sigma$$ as in § 3.2. Recall that $$\lambda$$ is the dominant cocharacter with $$\tau \in t^\lambda W_0$$. We will associate to $$\sigma$$ a superbasic element for a parabolic subgroup of $$\tilde W$$ and reduce Theorem 2.1 (2) to the superbasic case. We follow the approach in [10, Section 5]. Let $$V^\sigma$$ be the fixed point set of $$\sigma$$. Since $$\sigma$$ is an affine transformation on $$V$$ of finite order, $$V^\sigma$$ is a nonempty affine subspace. Set $$V'=\{v-e; v \in V^\sigma\}$$, where $$e$$ is an arbitrary point of $$V^\sigma$$. Then $$V'$$ is the (linear) subspace of $$V$$ parallel to $$V^\sigma$$. We choose a generic point $$v_0$$ of $$V'$$, that is for any root $$\alpha \in R$$, $$\langle\alpha, v_0\rangle=0$$ implies that $$\langle\alpha, v'\rangle=0$$ for all $$v' \in V'$$. We set $$I=I(\bar v_0)$$, $$J=J(\bar v_0)$$ and $$\sigma^J=z \sigma z^{-1} \in \tilde W \sigma$$, where $$z$$ is the unique element in $${}^J W_0$$ with $$\bar v_0=z(v_0)$$. Lemma 4.1. (1) The set $$J$$ is stable under $$\sigma_0$$-conjugation. (2) $$z(\lambda)^\clubsuit_{\sigma_0} \in \mathbb Q R_J^\vee$$. (3) The element $$\sigma^J$$ is a superbasic element for $$\tilde W_J$$. □ Proof (1) By definition, $$\sigma(0)=\lambda$$. Hence $$\sigma(v_0)=v_0+\lambda$$ and $$\sigma^J(\bar v_0)=\bar v_0+z(\lambda)$$. Write $$\sigma^J$$ as $$\sigma^J=t^{z(\lambda)}u \sigma_0$$ for some $$u \in W_0$$. Then $$u\sigma_0(\bar v_0)=\bar v_0$$. Therefore $$\sigma_0(\bar v_0)=u^{-1}(\bar v_0)$$ is the unique dominant element in the $$W_0$$-orbit of $$v_0$$. Hence $$\bar v_0=\sigma_0(\bar v_0)=u^{-1}(\bar v_0)$$. Therefore $$u^{-1}n W_J$$ and $$\text{Ad}(\sigma_0)(J)=J$$. (2) Since $$\sigma^J$$ is of finite order, we have $$(\sigma^J)^m=1$$ for some $$m \in \mathbb N$$. On the other hand, using the expression $$\sigma^J=t^{z(\lambda)}u \sigma_0$$, one computes that $$(\sigma^J)^m=t^{\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda))}=1$$. So $$\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda))=0$$. Since $$u^{-1}n W_J$$ and $$\sigma_0(R_J^\vee)=R_J^\vee$$, we have $$(u\sigma_0)^k(z(\lambda))-\sigma_0^k(z(\lambda)) \in \mathbb Z R_J^\vee$$ for $$k \in \mathbb Z$$. Thus $$(z(\lambda))^\clubsuit_{\sigma_0} \in \frac{1}{m}\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda)) + \mathbb Q R_J^\vee=\mathbb Q R_J^\vee$$. (3) Since $$\sigma_0$$ stabilizes $$\mathbf a_J$$, the length function $$\ell_J$$ on $$\tilde W_J$$ extends to the subgroup of $$\text{Aff}(V)$$ generated by $$\tilde W_J$$ and $$\sigma_0$$ via the usual rule $$\ell_J(w \sigma_0^i)=\ell_J(w)$$ for $$w \in \tilde W_J$$ and $$i \in \mathbb Z$$. Since $$z^{-1}(R_J^+) \subseteq R^+$$, we have $$z(\mathbf a) \subseteq \mathbf a_J$$. In other words, $$\mathbf a_J$$ is the unique alcove associated to $$\tilde W_J$$ that contains $$z(\mathbf a)$$. Since $$\sigma^J(z(\mathbf a))=z(\mathbf a)$$, $$\sigma^J(\mathbf a_J)$$ is also the unique alcove associated to $$\tilde W_J$$ that contains $$z(\mathbf a)$$. Therefore $$\sigma^J(\mathbf a_J)=\mathbf a_J$$. Since $$v_0$$ is generic in $$V'$$, $$\bar v_0=z(v_0)$$ is generic in $$z(V')$$. So each point of $$z(V')$$ is fixed by $$W_J$$. Therefore, for any $$\tilde \alpha \in \tilde R_J$$, either $$V^{\sigma^J} \cap H_{\tilde \alpha} = \emptyset$$ or $$V^{\sigma^J} \subseteq H_{\tilde \alpha}$$, where $$V^{\sigma^J}=z(V^\sigma)$$ is the fixed-point set of $$\sigma^J$$ on $$V$$. Since $$\sigma^J(\mathbf a_J)=\mathbf a_J$$ and $$\sigma^J$$ is of finite order, $$\mathbf a_J$$ contains a fixed point of $$\sigma^J$$. So $$V^{\sigma^J} \nsubseteq H_{\tilde \alpha}$$ and hence $$V^{\sigma^J} \cap H_{\tilde \alpha}=\emptyset$$. By [9, Proposition 3.5] (for $$\tilde W_G:=\tilde W_J, J_\mathcal O:=J$$ and $$y=1$$), $$\sigma^J$$ is superbasic for $$\tilde W_J$$. ■ Lemma 4.2. Let $$c$$ be a $$\sigma_0$$-orbit of $$\mathbb S$$. Then $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$ if $$c \subseteq I$$. □ Proof Write $$\lambda=z(\lambda)+\theta$$ for some $$\theta \in \mathbb Z R^\vee$$. We have   ⟨ωc,λσ0♣⟩=⟨ωc,z(λ)σ0♣⟩+⟨ωc,θ⟩≡⟨ωc,z(λ)σ0♣⟩modZ. By Lemma 4.1 (2), $$\langle\omega_c, z(\lambda)^\clubsuit_{\sigma_0}\rangle=0$$ if $$c \subseteq I$$. The proof is finished. ■ Proposition 4.3. The maximal Newton point of $$B(\tilde W, \mu, \sigma)$$ is contained in the natural inclusion $$B(\tilde W_J, \mu, \sigma^J) \hookrightarrow B(\tilde W, \mu, \sigma)$$. □ Proof For any $$j \in J$$, we denote by $$\omega_j^J$$ the fundamental weight corresponding to $$j$$ in the root datum $$\mathfrak R_J$$. We set $$\omega_c^J=\sum_{j \in c} \omega_j^J$$ for any $$\sigma_0$$-orbit of $$c$$ of $$J$$. Let $$\nu$$ be the maximal Newton point of $$B(\tilde W, \mu, \sigma)$$. Let $$c$$ be a $$\sigma_0$$-orbit of $$I$$. By Lemma 4.2, $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$. Applying Lemma 3.5 (a), we see that $$\langle\omega_c, \mu^\clubsuit_{\sigma_0}\rangle=\langle\omega_c, \nu\rangle$$ and $$\mu^\clubsuit_{\sigma_0} - \nu^{-1}n \mathbb Q R_J^\vee$$. By Lemma 4.1 (2), $$z(\lambda)^\clubsuit_{\sigma_0} \in \mathbb Q R_J^\vee$$. Thus $$\mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu^{-1}n \mathbb Q R_J^\vee$$. Now let $$c'$$ be a $$\sigma_0$$-orbit in $$I(\nu) \cap J$$. Then   ⟨ωc′J,μσ0♣+z(λ)σ0♣−ν⟩=⟨ωc′,μσ0♣+z(λ)σ0♣−ν⟩=⟨ωc′,μσ0♣+λσ0♣−ν⟩−⟨ωc′,θ⟩, where $$\theta=\lambda-z(\lambda) \in \mathbb Z R^\vee$$. By Lemma 3.5 $$\langle\omega_{c'}, \mu^\clubsuit_{\sigma_0}+\lambda^\clubsuit_{\sigma_0}-\nu\rangle \in \mathbb Z$$. Hence $$\langle\omega_{c'}^J, \mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu\rangle \in \mathbb Z$$. Again by Lemma 3.5, we have $$\pi_J(\nu) \in B((\tilde W_J)_{ad}, \pi_J(\mu), (\sigma^J)_{ad})$$, where $$\pi_J$$ and (respectively $$(\sigma^J)_{ad}$$) is defined similarly as $$\pi$$ (respectively $$\sigma_{ad}$$) in § 3.1 with $$\mathfrak R$$ and $$\sigma$$ replaced by $$\mathfrak R_J$$ and $$\sigma^J$$ , respectively. Since $$\mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu^{-1}n \mathbb Q R_J^\vee$$ and   πJ:B(W~J,μ,σJ)→B((W~J)ad,πJ(μ),(σJ)ad) is a bijection of posets, we deduce that $$\nu^{-1}n B(\tilde W_J, \mu, \sigma^J)$$ as desired. ■ Lemma 4.4. Let $$K \subseteq \mathbb S$$ and $$z \in {}^K W_0$$. If $$w, w' \in \tilde W_K$$ with $$w \le_K w'$$ for the Bruhat order of $$\tilde W_K$$, then $$z^{-1} w z \le z^{-1} w' z$$ for the Bruhat order of $$\tilde W$$. □ Proof By the definition of Bruhat order, there exist positive affine roots $$\tilde \alpha_1, \cdots, \tilde \alpha_k$$ of $$\tilde W_K$$ such that   w≤Kwsα~1≤Kwsα~1sα~2≤K⋯≤Kwsα~1⋯sα~k=w′. Hence for any $$i$$, $$w s_{\tilde \alpha_1} \cdots s_{\tilde \alpha_i}(\tilde \alpha_{i+1})$$ is a positive affine root of $$\tilde W_K$$. Notice that $$z^{-1}$$ sends positive affine roots of $$\tilde W_K$$ to positive affine roots of $$\tilde W$$. Set $$\tilde \beta_i=z^{-1}(\tilde \alpha_i)$$. This is a positive affine root of $$\tilde W$$. Moreover, $$(z^{-1} w z) s_{\tilde \beta_1} \cdots s_{\tilde \beta_i}(\tilde \beta_{i+1})=z^{-1} w s_{\tilde \alpha_1} \cdots s_{\tilde \alpha_i} (\tilde \alpha_{i+1})$$ is a positive affine root of $$\tilde W$$. Thus   z−1wz≤z−1wzsβ~1≤z−1wzsβ~1sβ~2≤⋯≤z−1wzsβ~1⋯sβ~k=z−1w′z. ■ Corollary 4.5. If Theorem 2.1 (2) holds for $$B(\tilde W_J, \mu, \sigma^J)$$, then it holds for $$B(\tilde W, \mu, \sigma)$$. □ Proof Let $$\nu$$ be the maximal Newton point of $$B(\tilde W, \mu, \sigma)$$, which is also the maximal Newton point of $$B(\tilde W_J, \mu, \sigma^J)$$ by Proposition 4.3. By assumption, there exist $$w_1 \in t^\mu (W_a \cap \tilde W_J)$$ and $$x_1 \in W_J$$ such that $$\bar \nu_{w_1, \sigma^J}^J=\nu$$ and $$w_1 \le_J t^{x_1(\mu)}$$, where $$\bar \nu^J_{w_1, \sigma^J}$$ stands for the Newton point of $$w_1 \in \tilde W_J$$ defined with respect to the $$\sigma^J$$-conjugation action on $$\tilde W_J$$. Let $$z$$ be the element defined in Section 4.3. Let $$w=z^{-1} w_1 z$$ and $$x=z^{-1} x_1$$. Then we have $$\bar \nu_{w, \sigma}=\nu$$, $$w \in t^\mu W_a$$ and $$w \le t^{x(\mu)}$$ as desired. ■ 5 Reduction to the Irreducible Case 5.1 In this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$ and $$\sigma$$ acts transitively on the set of connected components of the affine Dynkin diagram of $$W_a$$. In other words, $$\tilde W=\tilde W_1 \times \cdots \times \tilde W_m$$, where $$\tilde W_1 \cong \cdots \cong \tilde W_m$$ are extended affine Weyl groups of adjoint type with connected affine Dynkin diagram and $$\text{Ad}(\sigma)(\tilde W_1)=\tilde W_2, \cdots, \text{Ad}(\sigma)(\tilde W_m)=\tilde W_1$$. Let $$W_i$$ be the finite Weyl group associated to $$\tilde W_i$$. As in Section 3.2, we write $$\sigma$$ as $$\sigma=\tau \sigma_0$$ with $$\tau \in \Omega$$ and $$\text{Ad}(\sigma_0)(\mathbb S)=\mathbb S$$. Write $$\mu$$ as $$\mu=(\mu_1, \cdots, \mu_m)$$, where each $$\mu_i$$ is a dominant cocharacter for $$\tilde W_i$$. Let $$y=(w_1, \dots, w_m) \in \tilde W$$. Then the $$m$$-th component of $$(y\sigma)^m \sigma^{-m} \in \tilde W$$ is $$w_m\cdots \text{Ad}(\sigma^{m-1})(w_1)$$. The map $$\bar \nu_{(w_1, \dots, w_m), \sigma} \mapsto \bar \nu_{w_m\cdots \text{Ad}(\sigma^{m-1})(w_1), \sigma^m}$$ induces a natural surjection from $$B(\tilde W, \mu, \sigma)$$ to $$B(\tilde W_m, \gamma, \sigma^m)$$, where $$\gamma=\sum_{i=1}^m \sigma_0^{m-i} (\mu_i)$$. Since the elements in $$B(\tilde W, \mu, \sigma)$$ are $$\sigma_0$$-invariant, it is in fact a bijection, whose inverse is given by $$v \mapsto \frac{1}{m}(\sigma_0(v), \sigma_0^2(v), \cdots, v)$$. It is easy to see this bijection is a bijection of posets. Lemma 5.1. If Theorem 2.1 (2) holds for $$(\tilde W_m, \gamma, \sigma^m)$$, then it holds for $$(\tilde W, \mu, \sigma)$$. □ Proof Let $$\nu$$ be the maximal element in $$B(\tilde W_m, \gamma, \sigma^m)$$. Then the maximal element in $$B(\tilde W, \mu, \sigma)$$ is $$\frac{1}{m}(\sigma_0(\nu), \cdots, \nu)$$. By assumption, there exists $$w \in \text{Adm}(\gamma)$$ such that $$\bar \nu_{w, \sigma^m}=\nu$$. By definition, there exists $$x \in W_m$$ such that $$w \le t^{x(\gamma)}$$. Since $$\ell(t^{x(\gamma)})=\sum_{i=1}^m \ell(t^{x(\sigma_0^{m-i}(\mu_i))})$$, there exists $$w_i \in \tilde W_m$$ for each $$i$$ such that $$w=w_m \cdots w_1$$ and $$w_i \le t^{x(\sigma_0^{m-i}(\mu_i))}$$ for all $$i$$. It is easy to see that $$\sigma^{i-m}=\tau'_i \sigma_0^{i-m}$$ for some $$\tau'_i \in \tilde W$$. Hence   Ad(σ)i−m(wi)≤Ad(σ)i−m(tx(σ0m−i(μi)))=Ad(τi′)Ad(σ0)i−m(tx(σ0m−i(μi)))=txi(μi) for some $$x_i \in W_i$$. Set $$y=(\text{Ad}(\sigma)^{1-m}(w_1), \cdots, w_m) \in \tilde W$$. Then $$y \in \text{Adm}(\mu)$$. Notice that the $$m$$-th component of $$(y \sigma)^m \sigma^{-m}$$ is $$w_m \cdots w_1=w$$. Hence $$\bar \nu_{y, \sigma}=\frac{1}{m}(\sigma_0(\nu), \cdots, \nu)$$. ■ 6 The Irreducible Superbasic Case 6.1 In this section, we consider the extended affine Weyl group $$\tilde W=\mathbb Z^n \rtimes \mathfrak{S}_n$$ of type $$\tilde A_{n-1}$$, where $$\mathfrak{S}_n$$ is the permutation group of $$\{1, 2, \dots, n\}$$ which acts on $$\mathbb Z^n \cong \oplus_{i=1}^n \mathbb Z e_i^\vee$$ by $$w(e_i^\vee)=e_{w(i)}^\vee$$ for $$w \in \mathfrak{S}_n$$. Let $$\{e_i\}_{i=1, \cdots, n}$$ be the dual basis. Set $$d=\sum_{i=1}^n e_i$$ and $$d^\vee=\sum_{i=1}^n e_i^\vee$$. The simple roots, fundamental weights, and fundamental coweights are given by $$\alpha_i=e_i-e_{i+1}$$, $$\omega_{i,n}=-\frac{i}{n}d+\sum_{j=1}^i e_j$$ and $$\omega_{i,n}^\vee=-\frac{i}{n}d^\vee+\sum_{j=1}^i e_j^\vee$$ , respectively for $$i$$ with $$1 \le i \le n-1$$. Then $$\tilde W_{ad}=(\oplus_{i=1}^{n-1} \mathbb Z \omega_{i, n}^\vee) \rtimes \mathfrak{S}_n$$, see Section 3.1. Set $$\varpi_{m,n}=\sum_{j=1}^m e_j^\vee$$. The map $$\pi$$ in Section 3.1 can be described explicitly as the $$\mathbb Q$$-linear projection $$\pi: \mathbb Q^n \to \mathbb Q R^\vee \subseteq \mathbb Q^n$$ such that $$d^\vee \mapsto 0$$ and $$\varpi_{m,n} \mapsto \omega_{m,n}^\vee$$ for $$m$$ with $$1 \le m \le n-1$$. We also denote the induced (surjective) projection $$\tilde W \to \tilde W_{ad}$$ by $$\pi$$. For any positive integer $$m<n$$, let $$\sigma_{m, n}=t^{\varpi_{m,n}} u_{m,n} \in t^{\varpi_{m,n}} \mathfrak{S}_n$$ be the unique length zero element with $$u_{m, n} \in \mathfrak{S}_n$$. Then any superbasic element in $$\tilde W_{ad}$$ is of the form $$\pi(\sigma_{m, n})$$ for some positive integer $$m<n$$ co-prime to $$n$$. The main purpose of this section is to prove the following result. proposition 6.1. Let $$m<n$$ be a positive integer co-prime to $$n$$. Let $$\mu \in \oplus_{i=1}^n \mathbb Z e_i^\vee$$ be a dominant cocharacter. Then there exist $$\tilde W \in \tilde W$$ and $$x \in \mathfrak{S}_n$$ such that $$\tilde W \le t^{x(\mu)}$$ and $$\pi(\bar \nu_{\tilde W, \sigma_{m,n}})=\bar \nu_{\pi(\tilde W), \pi(\sigma_{m, n})}$$ equals the unique maximal Newton point $$\nu$$ of $$B(\tilde W_{ad}, \pi(\mu), \pi(\sigma_{m,n}))$$. □ The proof will be given in Section 6.6. 6.2 We first show that Proposition 6.1 implies Theorem 2.1 (2) for any triple $$(\tilde W_1, \mu_1, \sigma_1)$$. Let $$\mathfrak R$$ be the root datum of $$\tilde W_1$$. By Section 3.1, we may assume $$\mathfrak R=\mathfrak R_{ad}$$. By Corollary 4.5, it suffices to prove Theorem 2.1 (2) for $$(\tilde W_2, \mu_2, \sigma_2)$$, where $$\sigma_2$$ is a superbasic element in $$\tilde W_2$$. By Section 3.1 again, we may assume that the root datum of $$\tilde W_2$$ is adjoint. By Section 4.1, we may assume furthermore that $$\tilde W_2=\tilde W_3^m$$, where $$\tilde W_3$$ is the extended affine Weyl group of an adjoint root datum of type $$A$$ and $$\sigma_2$$ acts transitively on the set of affine simple reflections of $$\tilde W_2$$. By Lemma 5.1, it suffices to prove Theorem 2.1 (2) for $$(\tilde W_3, \mu_3, \sigma_3)$$, where $$\sigma_3=\sigma_2^m$$ is a superbasic element in $$\tilde W_3$$. This case follows from Proposition 6.1. 6.3 We recall the definition of $$\underline {\mathbf a}$$-sequence and $$\chi_{m, n}$$ in [6, Sections 3 and 5]. For $$i, j \in \mathbb Z$$, we set $$[i, j]=\{k \in \mathbb Z; i \le k \le j\}$$. Let $$r \in \mathbb N$$ and $$\chi \in \mathbb Z^r$$. For each $$j \in [1, r]$$ we define $$\underline {\mathbf a}_\chi^j: \mathbb Z_{\geq 0} \to \mathbb Z$$ by $$\underline {\mathbf a}_\chi^j(k)=\chi(j-k)$$. Here we identify $$l$$ with $$l+r$$ for $$l \in \mathbb Z$$. We say $$i \geq_\chi j$$ if $$\underline {\mathbf a}_\chi^i \geq \underline {\mathbf a}_\chi^j$$ in the sense of lexicographic order. If $$\ge_\chi$$ is a linear order, we define $$\epsilon_\chi \in \mathfrak{S}_r$$ such that $$\epsilon_\chi(i) < \epsilon_\chi(j)$$ if and only if $$i >_\chi j$$. Let $$s \leq r$$ be two nonnegative integers which are co-prime. Define $$\chi_{s, r} \in \mathbb Z^r$$ by $$\chi_{s, r}(i)=\lfloor \frac{is}{r} \rfloor-\lfloor (i-1) \frac{s}{r} \rfloor$$ for $$i \in [1, r]$$. Set $$\epsilon_{s, r}=\epsilon_{\chi_{s, r}}$$, which is well defined since $$s$$ and $$r$$ are co-prime. The following explicit description of $$\geq_{\chi_{m,n}}$$ and its application to the proof of Lemma 6.2 below are kindly suggested by one of the anonymous referees. First we identify $$[1, n]$$ with $$\mathbb Z / n \mathbb Z$$ in the natural way. Then one checks by definition that   χm,n(i)={1, if mi∈[0,m−1]⊆Z/nZ0, otherwise.  (a) Since $$m$$ is co-prime to $$n$$, $$m$$ is invertible in $$\mathbb Z / n \mathbb Z$$. We claim that   n=m−1⋅0>χm,nm−1⋅1>χm,n⋯>χm,nm−1⋅(n−1). (b) Indeed, if $$m^{-1} \cdot j_0 <_{\chi_{m,n}} m^{-1} \cdot (j_0+1)$$ for some $$j_0 \in [0, n-2]$$, there exists $$0 \leq k \leq n-1$$ such that $$\chi_{m,n}(m^{-1} \cdot j_0-i) = \chi_{m,n}(m^{-1} \cdot (j_0+1)-i)$$ for $$0 \leq i \leq k-1$$ and $$\chi_{m,n}(m^{-1} \cdot j_0-k) < \chi_{m,n}(m^{-1} \cdot (j_0+1)-k)$$. In particular, we have $$j_0+1-km=0 \in \mathbb Z / n\mathbb Z$$ by (a). Thus $$k \geq 1$$ and   χm,n(m−1⋅j0−k+1)=1>0=χm,n(m−1⋅(j0+1)−k+1), which is a contradiction. So (b) is proved. Now it is easy to see that $$\epsilon_{m,n}$$ is the permutation on $$[1, n]=\mathbb Z / n\mathbb Z$$ given by   ϵm,n(i)=mi+1 for i∈[1,n]. (c) 6.4 Let $$\mathcal S=\cup_{1 \le i \le j}\mathbb Z^{[i, j]}$$, whose elements are called segments. Let $$\eta \in \mathcal S$$ be a segment. Assume $$\eta \in \mathbb Z^{[i, j]}$$. We call $$\text{h}(\eta)=i$$ and $$\tilde \tau(\eta)=j$$ the head and the tail of $$\eta$$ , respectively. We call the positive integer $$j-i+1$$ the size of $$\eta$$. We set   |η|=∑k=h(η)t(η)η(k),av(η)=1t(η)−h(η)+1|η|. Let $$[i', j'] \subseteq [i, j]$$ be a sub-interval, we call the restriction $$\eta|_{[i',j']}$$ of $$\eta$$ to $$[i', j']$$ a subsegment of $$\eta$$ and write $$\eta|_i=\eta|_{[i,i]}$$. Let $$\theta$$ be an another segment such that $$\text{h}(\theta)=\tilde \tau(\eta)+1$$. We denote by $$\eta \vee \theta \in \mathbb Z^{[\text{h}(\eta), \tilde \tau(\theta)]}$$ the natural concatenation of $$\eta$$ and $$\theta$$. For $$k \in \mathbb Z$$, we denote by $$\eta[k]$$ the $$k$$-shift of $$\eta$$ defined by $$\eta[k](i)=\eta(i+k)$$. We say two segments are of the same type if they can be identified with each other up to some shift. For $$\eta \in \mathbb Q^{[1,n]}$$ , we denote by $$\text{Con}(\eta) \in \mathbb R^2$$ the convex hull of the points $$(0,0)$$ and $$(k, |\eta|_{[1, k]}|)$$ for $$k \in [1,n]$$. We say a subsegment $$\gamma$$ of $$\eta$$ is sharp if   av(γ)=max{av(γ′);γ′ is a subsegment of η with h(γ′)=h(γ)} and   av(γ)=min{av(γ′);γ′ is a subsegment of η with t(γ′)=t(γ)}. If $$\eta=\gamma^1 \vee \gamma^2 \vee \cdots \vee \gamma^s$$ with each $$\gamma^k$$ a sharp subsegment, then the points $$(0,0)$$ and $$(\tilde \tau(\gamma^i), |\gamma^1 \vee \cdots \vee \gamma^i|)$$ in $$\mathbb R^2$$ for $$i \in [1,s]$$ lie on the boundary of $$\text{Con}(\eta)$$ and their convex hull is just $$\text{Con}(\eta)$$. We call the dominant vector   sl(Con(η))=(av(γ1)∨⋯∨av(γs))∈Q[1,n] the slope sequence of $$\text{Con}(\eta)$$. Here for any $$\gamma \in \mathcal S$$, we define $$\text{av}^{(\gamma)} \in \mathbb Q^{[\text{h}(\gamma),\tilde \tau(\gamma)]}$$ by $$\text{av}^{(\gamma)}(i)=\text{av}(\gamma)$$ for $$i \in [\text{h}(\gamma),\tilde \tau(\gamma)]$$. Let $$\mu \in \mathbb Z^{[1,n]}$$ be a dominant cocharacter. Set $$\mu_{m,n}=\mu+\chi_{m,n}$$. Then   ⟨ωi,n,π(μm,n)⟩=⟨ωi,n,π(μ)⟩−(min−⌊min⌋)=⟨ωi,n,π(μ)+π(ϖm,n)⟩−⌈⟨ωi,n,π(ϖm,n)⟩⌉. According to the proof of Theorem 2.1 (1), the slope sequence   ν=sl(Con(π(μm,n)))=π(sl(Con(μm,n)) is the unique maximal Newton point of $$B(\tilde W_{ad}, \pi(\mu), \pi(\sigma_{m,n}))$$. 6.5 Similar to [6, Section 5], we use the Euclidean algorithm to give a recursive construction of $$\chi_{m,n}$$, which plays a crucial role in the proof of Proposition 6.1. Let $$D=\{(m,n) \in \mathbb Z_{>0}^2; m<n \text{ are co-prime}\}$$. We define $$f: D \to D \sqcup \{(1,1), (0,1)\}$$ by   f(m,n)={(m(⌊nm⌋+1)−n,m), if nm≥2;(n−(n−m)⌊nn−m⌋,n−m), otherwise. Define two types of segments $$1_{m,n}$$ and $$0_{m,n}$$ by   1m,n={(0(⌊nm⌋−1),1), if nm≥2;(0,1(⌊nn−m⌋)), otherwise,0m,n={(0(⌊nm⌋),1), if nm≥2;(0,1(⌊nn−m⌋−1)), otherwise, where the superscript $$^{(k)}$$ means to repeat the entry $$k$$ times. Here we do not fix the head or tail of these segments yet, as we are going to apply various shifts to them below. Set $$\mathcal S_1=\{\eta \in \mathcal S; \eta(i) \in \{0, 1\} \text{ for } i \in [\text{h}(\eta), \tilde \tau(\eta)]\}$$. For $$\eta \in \mathcal S_1$$ and $$k \in [\text{h}(\eta), \tilde \tau(\eta)]$$, set   η(k)m,n={1m,n, if η(k)=1;0m,n, if η(k)=0. For $$k \in [\text{h}(\eta), \tilde \tau(\eta)]$$, let $$\eta_{m,n,k}$$ be a shift of $$\eta(k)_{m,n}$$ whose head is determined recursively as follows:   h(ηm,n,k)={h(η), if k=h(η);t(ηm,n,k−1)+1, if k>h(η). Now we define $$\phi_{m,n}: \mathcal S_1 \to \mathcal S_1$$ by $$\phi_{m,n}(\eta)=\eta_{m,n,\text{h}(\eta)} \vee \cdots \vee \eta_{m,n,\tilde \tau(\eta)}$$ for $$\eta \in \mathcal S_1$$. If $$f^{h-1}(m,n) \in D$$, we set $$\phi_{m,n,h}=\phi_{m,n} \circ \cdots \circ \phi_{f^{h-1}(m,n)}$$. Using the Euclidean algorithm, one checks that   ϕm,n,h(χfh(m,n))=χm,n. We say a subsegment $$\gamma$$ of $$\chi_{m,n}$$ is of level$$h$$ if it is the image of some subsegment $$\gamma^h$$ of $$\chi_{f^h(m,n)}$$ under the map $$\phi_{m,n,h}$$. When $$h=1$$ and $$\gamma^h$$ is of size one, we say $$\gamma$$ is an elementary subsegment of $$\chi_{m,n}$$. Let $$\beta^1$$ and $$\gamma^1$$ be two segments of $$\chi^1=\chi_{f(m,n)}$$ and let $$\gamma$$ be a level one subsegment of $$\chi=\chi_{m,n}$$. Using the Euclidean algorithm, we have the following basic facts: (a) $$\text{av}(\beta^1) \geq \text{av}(\gamma^1)$$ if and only if $$\text{av}(\phi_{m,n}(\beta^1)) \geq \text{av}(\phi_{m,n}(\gamma^1))$$. (b) Each sharp subsegment of $$\gamma$$ with the same head is of level one. (c) If, moreover, $$\gamma$$ is an elementary subsegment of $$\chi$$, then $$\underline {\mathbf a}_\chi^{j} < \underline {\mathbf a}_\chi^{\text{h}(\gamma)-1}$$ and $$\underline {\mathbf a}_\chi^{j} < \underline {\mathbf a}_\chi^{\tilde \tau(\gamma)}$$ for $$j \in [\text{h}(\gamma), \tilde \tau(\gamma)-1]$$. (d) $$\underline {\mathbf a}_{\chi^1}^i < \underline {\mathbf a}_{\chi^1}^j$$ if and only if $$\underline {\mathbf a}_\chi^{\tilde \tau(\phi_{m,n}(\chi^1 |_i))} < \underline {\mathbf a}_\chi^{\tilde \tau(\phi_{m,n}(\chi^1 |_j))}$$. (e) $$\epsilon_{m,n}(n)=1$$. 6.6 Proof of Proposition 6.1 For a sequence of (distinct) elements $$i_1, i_2, \dots, i_r$$ in $$[1, n]$$, we denote by $$\text{cyc}(i_1, i_2, \dots, i_r) \in \mathfrak{S}_n$$ the cyclic permutation $$i_1 \mapsto i_2 \mapsto \cdots \mapsto i_r \mapsto i_1$$, which acts trivially on the remaining elements of $$[1,n]$$. For $$\eta \in \mathcal S$$ we set   xη=cyc(h(η),h(η)+1,…,t(η))∈S=∪i=1∞Si. Similarly, for a sequence $$\textbf{c}=(c^1, \dots, c^s)$$ of segments, we set $$x_{\textbf{c}}=x_{c^1, \dots, c^s}=x_{c^1} \cdots x_{c^s}$$. If $$\eta=c^1 \vee \cdots \vee c^s$$, we say $$\textbf{c}$$ is a decomposition of $$\eta$$. Now we are ready to prove Proposition 6.1. Write $$\chi=\chi_{m,n}$$, $$\theta=\mu_{m,n}=\mu+\chi$$ and $$\epsilon_=\epsilon_{m,n}$$. For $$h \in \mathbb Z_{>0}$$ we set $$\phi_h=\phi_{m,n,h}$$ and $$\chi^h=\chi_{f^h(m,n)}$$. By § 6.4, we have $$\nu=\pi(\text{sl}(\text{Con}(\theta)))$$. The proof will proceed as follows. First we construct a suitable sharp decomposition $$\textbf{c}$$ of $$\theta$$. One checks directly $$\pi(\nu_{w_{\textbf{c}}, \text{id}})=\pi(\text{sl}(\text{Con}(\theta)))=\nu$$, where $$w_{\textbf{c}}=t^\theta x_{\textbf{c}} \in \tilde W$$. Then we show that   ϵwcϵ−1≤ϵtθxθϵ−1=tϵ(μ)σm,n, where the last equality follows from Lemma 6.2 below. Set $$\tilde W=\epsilon w_{\textbf{c}} \epsilon^{-1} \sigma_{m, n}^{-1}$$. Then $$\tilde W \,{\le}\, t^{\epsilon(\mu)}$$ and $$\pi(\nu_{\tilde W, \sigma_{m, n}})\,{=}\,\pi(\nu_{\epsilon w_{\textbf{c}} \epsilon^{-1}, id})\,{=}\,\epsilon(\nu)$$. This completes the proof of Proposition 6.1. Lemma 6.2. We have $$\sigma_{m,n}=\epsilon t^\chi x_\chi \epsilon^{-1}$$. □ Proof We use the explicit descriptions of $$\geq_\chi$$ and $$\epsilon$$ in § 6.3 via the identification of $$[1, n]$$ with $$\mathbb Z / n\mathbb Z$$ in the natural way. Then $${u_{1,n}}=x_\chi$$ is the permutation $$i \mapsto i+1$$ on $$\mathbb Z / n\mathbb Z$$. Since $$\sigma_{m,n}=\sigma_{1,n}^m$$, we have $$u_{m,n}(i)=i+m$$ for $$i \in \mathbb Z / n\mathbb Z$$. On the other hand, we already know that $$\epsilon(i)=mi+1$$ for $$i \in \mathbb Z / n\mathbb Z$$. Therefore, $$\epsilon x_\chi \epsilon^{-1}$$ is the permutation $$mi+1 \mapsto m(i+1)+1$$ on $$\mathbb Z / n\mathbb Z$$, which equals $$u_{m,n}$$ as desired. It remains to show $$\epsilon(\chi)=\varpi_{m,n}$$. For $$1 \leq i < j \leq n$$ we have $$\epsilon(\chi)(i)=\chi(m^{-1} \cdot (i-1)) \geq \chi(m^{-1} \cdot (j-1))=\epsilon(\chi)(j)$$ since $$m^{-1} \cdot (i-1) >_{\chi} m^{-1} \cdot (j-1)$$. Thus $$\epsilon(\chi)$$ is dominant and equals $$\varpi_{m,n}$$. The proof is finished. ■ Assume $$I(\mu)=\{j \in [1,n-1]; \langle\alpha_j, \mu\rangle \neq 0\}=\{b_1, b_2,\dots, b_{r-1}\}$$ with $$b_1 < b_2 < \cdots < b_{r-1}$$. We set $$b_0=0$$ and $$b_r=n$$. Set $$\theta^i=\theta|_{[b_{i-1}+1,b_i]}$$ for $$i \in [1,r]$$. Then $$\theta=\theta^1 \vee \cdots \vee \theta^r$$. Suppose we have a sharp decomposition $$\textbf{c}_i$$ of $$\theta^i$$ for $$i \in [1, r]$$. Since $$\chi \in \{0,1\}^{[1,n]}$$ and $$\theta=\mu+\chi$$, for any subsegment $$\eta^i$$ (respectively $$\eta^j$$) of $$\theta^i$$ (respectively $$\theta^j$$) we have $$\text{av}(\eta^i) \geq \text{av}(\eta^j)$$ if $$i<j$$. Therefore the natural union $$\textbf{c}=\textbf{c}_1 \vee \cdots \vee \textbf{c}_r$$ forms a sharp decomposition of $$\theta$$. Let $$1 \leq i \leq r$$. We will construct inductively the subsegments $$\zeta_i^j, \gamma_i^j, \xi_i^j$$ for $$j \in [1, l_i]$$ (some of them might be empty) such that (a) $$\gamma^0_i=\chi |_{[b_{i-1}+1, b_j]}$$ and $$\gamma_i^{j-1}=\zeta_i^j \vee \gamma_i^j \vee \xi_i^j$$ for $$j \in [1, l_i]$$; (b) $$\zeta_i^j$$ and $$\xi_i^j$$ are sharp subsegments of $$\gamma_i^{j-1}$$; any sharp subsegment of $$\gamma_i^j$$ is also a sharp subsegment of $$\gamma_i^{j-1}$$; $$\gamma_i^{l_i}$$ is a sharp subsegment of itself (self-sharp); (c) For any$$j$$, $$\epsilon z_{i,j-1} \epsilon^{-1} \ge \epsilon z_{i,j} \epsilon^{-1}$$. Here   zi,j=tθyi−1xijvi,j;yi=xc1⋯xci;xij=xζi1,…,ζij,ξij,…,ξi1;vi,j=xγijxθi+1∨⋯∨θrcyc(t(γij),n))=cyc(h(γij),…,t(γij),bi+1,…,n). Assume we have (a), (b), and (c) for all $$i$$ and $$j$$. Set $${\zeta'}_i^j=\theta |_{[\text{h}(\zeta_i^j),\tilde \tau(\zeta_i^j)]}$$, $${\gamma'}_i^{l_i}=\theta |_{[\text{h}(\gamma_i^{l_i}),\tilde \tau(\gamma_i^{l_i})]}$$ and $${\xi'}_i^j=\theta |_{[\text{h}(\xi_i^j), \tilde \tau(\xi_i^j)]}$$. Then   ci=(ζ′i1,ζ′i2,…,ζ′ili,γ′ili,ξ′ili,…,ξ′i2,ξ′i1) forms a sharp decomposition of $$\theta^i$$, and   ϵtθxθϵ−1=ϵz1,0ϵ−1≥⋯≥ϵz1,l1+1ϵ−1=ϵz2,0ϵ−1≥⋯≥ϵzr,lr+1ϵ−1=ϵwcϵ−1 as desired. The construction is as follows. Suppose for $$1 \leq k < i$$ and $$0 \leq l \leq j$$, $$\textbf{c}_k$$, $$z_i^l$$, $$\xi_i^l$$, $$\gamma_i^l$$ are already constructed, and moreover $$\epsilon z_{i,j-1} \epsilon^{-1} \ge \epsilon z_{i,j} \epsilon^{-1}$$. We construct $$\zeta_i^{j+1}, \gamma_i^{j+1}, \xi_i^{j+1}$$ and show that $$\epsilon z_{i,j} \epsilon^{-1} \ge \epsilon z_{i, j+1} \epsilon^{-1}$$. If $$\gamma_i^j$$ is empty, there is nothing to do. Otherwise, we assume $$\gamma_i^j$$ is of level $$h$$ but not of level $$h+1$$. Then $$\gamma_i^j=\phi_h(\iota)$$ for some subsegment $$\iota$$ of $$\chi^h$$. Case (I): $$\iota$$ is not a subsegment of any elementary subsegment of $$\chi^h$$. Then there exist unique subsegments $$\zeta$$, $$\gamma$$ and $$\xi$$ of $$\chi^h$$ such that $$\gamma$$ is of level one, $$\zeta$$ (respectively $$\xi$$) is a proper subsegment of some elementary segment of $$\chi^h$$ with the same tail (respectively head), and $$\iota=\zeta \vee \gamma \vee \xi$$. Notice that at least two of $$\zeta$$, $$\gamma$$ and $$\xi$$ are nonempty. Define $$\zeta_i^{j+1}=\phi_h(\zeta)$$, $$\gamma_i^{j+1}=\phi_h(\gamma)$$ and $$\xi_i^{j+1}=\phi_h(\xi)$$. Note that $$\text{av}(\chi^h|_{[\text{h}(\zeta), \tilde \tau(\zeta)]})$$ is maximal among all subsegments of $$\chi^h$$ with the same head and $$\text{av}(\chi^h|_{[\text{h}(\xi), \tilde \tau(\xi)]})$$ is minimal among all subsegments of $$\chi^h$$ with the same tail. Therefore, (b) follows from Section 6.5 (a) and (b). To prove (c), it suffices to show that   ϵzi,j+1ϵ−1≤ϵ zi,j cyc(n,t(ζij+1)) ϵ−1; (d)  ϵ zi,j cyc(n,t(ζij+1)) ϵ−1≤ϵzi,jϵ−1. (e) Note that   ϵzi,j+1ϵ−1={ϵ zi,j cyc(n,t(ζij+1)) cyc(h(ξij+1)−1,t(ξij+1)) ϵ−1, if γ≠∅;ϵ zi,j cyc(n,t(ζij+1)) cyc(n,t(ξij+1)) ϵ−1, otherwise. Here we take $$\text{cyc}(n, \tilde \tau(\zeta_i^{j+1}))$$ (respectively $$\text{cyc}(\text{h}(\xi_i^{j+1})-1, \tilde \tau(\xi_i^{j+1}))$$ and $$\text{cyc}(n, \tilde \tau(\xi_i^{j+1}))$$) to be the identity element of $$\mathfrak{S}_n$$ if $$\zeta$$ (respectively $$\xi$$) is empty, in which case the inequality (e) (respectively (d)) becomes a priori an equality. Now we prove (d). We suppose $$\xi$$ is nonempty. Otherwise, there is nothing to prove. First we assume $$\gamma \neq \emptyset$$. By § 6.5 (c), we have $$\underline {\mathbf a}_{\chi^h}^{\text{h}(\xi)-1} > \underline {\mathbf a}_{\chi^h}^{\tilde \tau(\xi)}$$. Hence by Section 6.5 (d), $$\epsilon(\text{h}(\xi_i^{j+1})-1) < \epsilon(\tilde \tau(\xi_i^{j+1}))$$ and $$\alpha=\epsilon(e_{\text{h}(\xi_i^{j+1})-1} - e_{\tilde \tau(\xi_i^{j+1})})$$ is a positive root. Then (d) is equivalent to the following inequality   ⟨α,ϵzi,j+1−1ϵ−1(a)⟩=⟨α,ϵ(vi,jxijyi−1)−1ϵ−1(a−ϵ(θ))⟩=⟨ϵ(vi,jxijyi−1)ϵ−1(α),a−ϵ(θ)⟩=−⟨ϵ(ebi+1−eh(ξij+1)),ϵ(θ)⟩+⟨ϵ(ebi+1−eh(ξij+1)),a⟩=θ(h(ξij+1))−θ(bi+1)+⟨ϵ(ebi+1−eh(ξij+1)),a⟩>0, where $$\mathbf a$$ is the base alcove defined in Section 2.2. Note that $$1 > \langle\epsilon(e_{b_i+1}-e_{\text{h}(\xi_i^{j+1})}), \mathbf a\rangle > -1$$. Therefore, to prove (d), we have to show either $$\theta(\text{h}(\xi_i^{j+1})) > \theta(b_i+1)$$ or $$\theta(\text{h}(\xi_i^{j+1})) = \theta(b_i+1)$$ and $$\epsilon(b_i+1) <\epsilon(\text{h}(\xi_i^{j+1}))$$. Note that we always have $$\theta(\text{h}(\xi_i^{j+1}))\geq \theta(b_i+1)$$. If $$\theta(\text{h}(\xi_i^{j+1})) = \theta(b_i+1)$$, then $$\chi(b_i+1)=1>0=\chi(\text{h}(\xi_i^{j+1}))$$. Hence $$\epsilon(b_i+1) <\epsilon(\text{h}(\xi_i^{j+1}))$$ as desired. Now we assume $$\gamma= \emptyset$$. Note that $$\epsilon(n) < \epsilon(\tilde \tau(\xi_i^{j+1}))$$. Then (d) follows by a similar argument as in the case of $$\gamma \neq \emptyset$$ with $$\text{h}(\xi_i^{j+1})-1$$ replaced by $$n$$. To prove (e), again we suppose that $$\zeta$$ is nonempty. Since one of $$\gamma$$ and $$\xi$$ is nonempty, $$\tilde \tau(\zeta_i^{j+1}) \neq n$$. By Section 6.5 (e), $$1=\epsilon(n) < \epsilon(\tilde \tau(\zeta_i^{j+1}))$$. As in the proof of (d), we see that (e) holds if $$\theta(\tilde \tau(\zeta_i^{j+1})+1) < \theta(\text{h}(\zeta_i^{j+1}))$$. Otherwise, we have $$\theta(\tilde \tau(\zeta_i^{j+1})+1) = \theta(\text{h}(\zeta_i^{j+1}))$$, which implies that $$\chi(\text{h}(\zeta_i^{j+1}))=\chi(\tilde \tau(\zeta_i^{j+1})+1)=0$$. Since $$\zeta$$ is a proper subsegment of some elementary segment of $$\chi^h$$ and shares the same tail with it, by Section 6.5 (c) we have that $$\underline {\mathbf a}_{\chi^h}^{\tilde \tau(\zeta)} > \underline {\mathbf a}_{\chi^h}^{\text{h}(\zeta)-1}$$. Hence by § 6.5 (d) and that $$\chi(\tilde \tau(\zeta_i^{j+1})+1)=\chi(\text{h}(\zeta_i^{j+1}))=0$$, we have $$\underline {\mathbf a}_\chi^{\tilde \tau(\zeta_i^{j+1})+1} > \underline {\mathbf a}_\chi^{\text{h}(\zeta_i^{j+1})}$$. So $$\epsilon(\text{h}(\zeta_i^{j+1})) > \epsilon(\tilde \tau(\zeta_i^{j+1})+1)$$ and (e) holds. Case (II): $$\iota$$ is a subsegment of some elementary subsegment of $$\chi^h$$. We define $$l_i=j$$ and the construction of $$\textbf{c}_i$$ is finished. One checks directly that $$\iota$$ is self-sharp, hence so is $$\gamma_i^{l_i}=\phi_h(\iota)$$ by Section 6.5 (a) and (b). If $$\tilde \tau(\gamma_i^{l_i})=n$$, the induction step is finished. Otherwise, it remains to show   ϵzi,liϵ−1≥ϵ zi,li cyc(t(γili),n) ϵ−1=ϵzi,li+1ϵ−1. (f) Note that $$1=\epsilon(n) < \epsilon(\tilde \tau(\gamma_i^{l_i}))$$. Again, we see that (f) holds if $$\theta(\text{h}(\gamma_i^{l_i})) > \theta(b_i+1)$$. Otherwise, we have $$\theta(\text{h}(\gamma_i^{l_i})) = \theta(b_i+1)$$, $$\chi(\text{h}(\gamma_i^{l_i}))=0$$ and $$\chi(b_i+1)=1$$ since $$b_i \in I(\mu)$$. Hence $$\epsilon(\text{h}(\gamma_i^{l_i})) > \epsilon(b_i+1)$$ and (f) still holds. Example 6.3. Finally we provide an example. Let $$n=8$$, $$m=5$$ and $$\mu=(1, 1, 1, 0, 0, 0, 0, 0)$$. Then $$\chi_{m,n}=(0, 1, 0, 1, 1, 0, 1, 1) \in \mathbb Z^8$$, $$\epsilon_{m,n}=\text{cyc}(1, 6, 7, 4, 5, 2, 3, 8)$$ and $$u_{m, n}=\text{cyc}(6, 3, 8, 5, 2, 7, 4, 1)$$. Note that $$I(\mu)=3$$. Applying the algorithm in the proof of Proposition 6.1, we obtain the following sharp decomposition:   μm,n=(1,2,1,1,1,0,1,1)=(1,2)∨(1)∨(1,1)∨(0,1,1). Hence $$\nu=(\frac{1}{2}, \frac{1}{2}, 0, 0, 0, -\frac{1}{3}, -\frac{1}{3}, -\frac{1}{3})$$. Moreover, one checks that   tϵm,n(μ)σm,n≥tϵm,n(μ)σm,ncyc(8,3)≥tϵm,n(μ)σm,ncyc(8,3)cyc(1,3)≥tϵm,n(μ)σm,ncyc(8,3)cyc(1,3)cyc(1,2). Set $$\tilde W=t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3) \text{cyc}(1, 3) \text{cyc}(1, 2) \sigma_{m, n}^{-1}$$. Then $$\tilde W \le t^{\epsilon_{m, n}(\mu)}$$ and $$\pi(\bar \nu_{\tilde W, \sigma_{m,n}})=\nu$$. This verifies Proposition 6.1 in this case. □ Acknowledgement We thank the referees for their careful reading of this paper and many useful comments. Appendix: The Set $$B(G, \{\mu\})$$ In the appendix, we discuss the relation between the set $$B(\tilde W, \mu, \sigma)$$ defined in Section 2.4 and the set $$B(G, \mu)$$ for $$p$$-adic groups. A.1 Recall that $$F$$ is a finite field extension of $$\mathbb Q_p$$, $$L$$ is the completion of the maximal unramified extension of $$F$$ and $$G$$ is a connected reductive algebraic group over $$F$$. We first discuss the Iwahori-Weyl group of $$G$$ over $$L$$. We follow [5]. Let $$S$$ be a maximal $$L$$-split torus that is defined over $$F$$ and let $$T$$ be its centralizer. Since $$G$$ is quasi-split over $$L$$, $$T$$ is a maximal torus. Let $$N$$ be the normalizer of $$T$$. The finite Weyl group associated to $$S$$ is $$W_0=N(L)/T(L).$$ The Iwahori–Weyl group associated to $$S$$ is $$\tilde W=N(L)/T(L)_1$$, where $$T(L)_1$$ denotes the unique Iwahori subgroup of $$T(L)$$. Let $$\Gamma=\text{Gal}(\bar L/L)$$. As in [5, p. 195] and [15, Section 4.2], one associates a reduced root system $$R$$ to $$(G, T)$$. Let $$V=X_*(T)_\Gamma \otimes \mathbb R=X_*(S) \otimes \mathbb R$$. We fix a $$\sigma$$-invariant alcove $$\mathbf a_G$$ in the apartment of $$S$$ along with a special vertex of $$\mathbf a_G$$. The special vertex allows us to identify $$V$$ with the apartment of $$S$$. The alcove $$\mathbf a_G$$ is contained in a unique (relative) Weyl chamber of $$V$$, which we call the dominant chamber. The hyperplanes in $$V$$ then give a reduced root system $$R$$. We have the semi-direct product   W~=X∗(T)Γ⋊W0. The group $$X_*(T)_\Gamma$$ is not torsion-free in general. However, by [4, §8.1], the torsion part $$X_*(T)_{\Gamma, tor}$$ lies in the center of $$\tilde W$$ and one may identify the extended affine Weyl group of $$\mathfrak R$$ with $$\tilde W/X_*(T)_{\Gamma, tor}$$. For our purpose, it suffices to consider the case where $$X_*(T)_\Gamma$$ is torsion-free. In this case, the root system $$R$$, together with the cocharacter group $$X_*(T)_\Gamma$$, defines a reduced datum $$\mathfrak R$$, and $$\tilde W$$ is the extended affine Weyl group of $$\mathfrak R$$ introduced in Section 2. The Iwahori-Weyl group $$\tilde W$$ contains the affine Weyl group $$W_a$$ as a normal subgroup and   W~=Wa⋊Ω, where $$\Omega \cong \pi_1(G)_{\Gamma}$$ is the normalizer of the alcove $$\mathbf a_G$$. The Bruhat order on $$W_a$$ extends in a natural way to $$\tilde W$$. The Frobenius morphism $$\sigma$$ induces an action on $$\tilde W$$, which we denote by $$\text{Ad}(\sigma)$$. A.2 Recall that $$\{\mu\}$$ is a geometric conjugacy class of cocharacters of $$G$$. We may regard $$\{\mu\}$$ as a conjugacy class in $$X_*(T)$$ under the absolute Weyl group. Following [15, Section 4.3], let $$\tilde \Lambda_{\{\mu\}} \subseteq \{\mu\}$$ be the subset of cocharacters which are $$B$$-dominant for some Borel subgroup $$B$$ defined over $$L$$ with $$B \supseteq T$$. Then $$\tilde \Lambda_{\{\mu\}}$$ is a single $$W_0$$-orbit. Let $$\Lambda_{\{\mu\}}$$ be the image of $$\tilde \Lambda_{\{\mu\}}$$ in $$X_*(T)_\Gamma$$. The $$\{\mu\}$$-admissible set is defined by   Adm({μ})={w∈W~;w≤tξ for some ξ∈Λ{μ}}. Let $$\tilde \mu$$ be the unique dominant cocharacter in $$\tilde\Lambda_{\{\mu\}}$$ and $$\mu$$ be its image in $$\Lambda_{\{\mu\}}$$. Then $$\Lambda_{\{\mu\}}=W_0 \cdot \mu$$ and $$\text{Adm}(\{\mu\})$$ equals $$\text{Adm}(\mu)$$ defined in § 2.4. (In the function field case, it is shown in [17, Remark 2.11] that $$\text{Adm}(\{\mu\})=\{w \in \tilde W; w \le t^\xi \text{ for some } \xi \in im\{\mu\} \subseteq X_*(T)_\Gamma\}$$. We do not need this result here. A.3 The set $$B(G)$$ of $$\sigma$$-conjugacy classes of $$G(L)$$ is classified by Kottwitz in [11] and [12]. For any $$b \in G(L)$$, we denote by $$[b]$$ the $$\sigma$$-conjugacy class of $$G(L)$$ that contains $$b$$. Let $$\Gamma_F=\text{Gal}(\bar L/F)$$ be the absolute Galois group of $$F$$. Let $$\kappa_G: B(G) \to \pi_1(G)_{\Gamma_F}$$ be the Kottwitz map [12, § 7]. This gives one invariant. Another invariant is given by the Newton map. To an element $$b \in G(L)$$, we associate its Newton point $$\bar \nu_b \in X_*(T)_{\Gamma} \otimes \mathbb R$$. By [12, Section 4.13], the map   B(G)→π1(G)ΓF×(X∗(T)Γ⊗R),b↦(κG(b),ν¯b) is injective. For any $$w \in \tilde W$$, we choose a representative in $$N(L)$$ and also write it as $$w$$. The map $$N(L) \to G(L)$$ induces a map $$\tilde W \to B(G)$$. By [7, Section 3] and [3, 2.4], this map is surjective. ([3, 2.4] is stated for adjoint groups, but the result for arbitrary groups holds by combining with the reduction argument in [3, 2.3].) The restrictions of the Kottwitz map and the Newton map on $$\tilde W \subseteq G(L)$$ are the maps $$\kappa_{\tilde W, \sigma}$$ and $$w \mapsto \bar \nu_{w, \sigma}$$ defined in § 2.3. A.4 In this section, we assume furthermore that $$G$$ is a quasi-split connected reductive group over $$F$$ and that $$H$$ is an inner form of $$G$$. We denote by $$\sigma_G$$ and $$\sigma_H$$ the Frobenius morphisms of $$G$$ and $$H$$ , respectively. Via the canonical isomorphism $$X_*(T)_{\Gamma} \otimes \mathbb R \cong (X_*(T) \otimes \mathbb R)^\Gamma$$, we may identify $$\mu^\diamondsuit_{\sigma_G}$$ in § 2.4 with $$[\Gamma_F: \Gamma_{F, \tilde \mu}]^{-1} \sum_{\tau \in \Gamma_F/\Gamma_{F, \tilde \mu}} \tau(\tilde \mu)$$ in [12, (6.1.1)], where $$\Gamma_{F, \tilde \mu}$$ is the isotropy group of $$\mu$$ in $$\Gamma_F$$. By Lemma 3.4, $$\mu^\diamondsuit_{\sigma_G}=\mu^{\diamondsuit}_{\sigma_H}$$. Let $$\mu^\sharp$$ be the image of $$\tilde \mu$$ under the natural map $$X_*(T) \to \pi_1(H)_{\Gamma_F}$$. Set   B(H,{μ})={[b]∈B(H);κH(b)=μ♯,ν¯b≤μσH♢}. Then we may identify $$B(H, \{\mu\})$$ with $$B(\tilde W, \mu, \sigma_H)$$. Theorem 2.1 may be reformulated as follows: The set $$B(H, \{\mu\})$$ contains a unique maximal element and this element is represented by an element in $$\text{Adm}(\{\mu\})$$. References [1] Bourbaki N. Groupes et algèbres de Lie, Ch. 4,5,6 . Paris: Hermann, 1968. [2] Chai C. “Newton polygon as lattice points.” American Journal of Mathematics  122, no. 5 ( 2000): 967– 90. Google Scholar CrossRef Search ADS   [3] Görtz U. He X. and Nie. S. “$$P$$-alcoves and nonemptiness of affine Deligne-Lusztig varieties.” Annales Scientifiques de l’École Normale SupÉrieure  48 ( 2015): 647– 65. [4] Haines T. and He. X. “Vertexwise criteria for admissibility of alcoves.” arXiv:1411.5450, to appear in The American Journal of Mathematics. [5] Haines T. and Rapoport M. “On parahoric subgroups.” Advances in Mathematics 219 ( 2008): 188– 98. [6] He. X. “Minimal length elements in conjugacy classes of extended affine Weyl groups.” arXiv: 1004.4040. [7] He X. “Geometric and homological properties of affine Deligne-Lusztig varieties.” Annals of Mathematics  179 ( 2014): 367– 404. Google Scholar CrossRef Search ADS   [8] He. X. “Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties.” arXiv:1408.5838, to appear in Annales Scientifiques de l’Ècole Normale Supérieure. [9] He X. and S. Nie. “Minimal length elements of extended affine Weyl group.” Compositio Mathematica  150 ( 2014): 1903– 27. Google Scholar CrossRef Search ADS   [10] He X. and S. Nie. “$$P$$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras.” Selecta Mathematica  21 ( 2015): 995– 1019. Google Scholar CrossRef Search ADS   [11] Kottwitz R. “Isocrystals with additional structure.” Compositio Mathematica  56 ( 1985): 201– 20. [12] Kottwitz R. “Isocrystals with additional structure. II.” Compositio Mathematica  109 ( 1997): 255– 339. Google Scholar CrossRef Search ADS   [13] Kottwitz R. and Rapoport. M. “Minuscule alcoves for $$GL_n$$ and $$GSp_{2n$$.” Selecta Mathematica  102 ( 2000): 403– 28. [14] Kottwitz R. and Rapoport M. “On the existence of F -crystals.” Commentarii Mathematici Helvetici  78 ( 2003): 153– 84. Google Scholar CrossRef Search ADS   [15] Pappas G. Rapoport M. and B. Smithling. Local Models of Shimura Varieties, I. Geometry and Combinatorics,  pp. 135– 217. Handbook of moduli vol. III, Advanced Lectures in Mathematics (ALM) vol. 26. Somerville, MA: Int. Press, 2013. [16] Rapoport M. “A guide to the reduction modulo $$p$$ of Shimura varieties.” Astérisque  298 ( 2005): 271– 318. [17] Richarz T. “Affine Grassmannians and geometric satake equivalences.” arXiv:1311.1008. [18] Springer T. A. “Regular elements of finite reflection groups.” Inventiones Mathematicae  25 ( 1974): 159– 98. Google Scholar CrossRef Search ADS   © The Author 2016. Published by Oxford University Press. All rights reserved. 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# On the Acceptable Elements

, Volume 2018 (3) – Feb 1, 2018
25 pages

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Publisher
Oxford University Press
ISSN
1073-7928
eISSN
1687-0247
D.O.I.
10.1093/imrn/rnw260
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### Abstract

Abstract In this article, we study the set $$B(G, \{\mu\})$$ of acceptable elements for any $$p$$-adic group $$G$$. We show that $$B(G, \{\mu\})$$ contains a unique maximal element and the maximal element is represented by an element in the admissible subset of the associated Iwahori–Weyl group. 1 Introduction Let $$F$$ be a finite field extension of $$\mathbb Q_p$$ and $$L$$ be the completion of the maximal unramified extension of $$F$$. Let $$G$$ be a connected reductive algebraic group over $$F$$ and $$\sigma$$ be the Frobenius morphism. We denote by $$B(G)$$ the set of $$\sigma$$-conjugacy classes of $$G(L)$$. The set $$B(G)$$ is classified by Kottwitz in [11] and [12]. This classification generalizes the Dieudonnè–Manin classification of isocrystals by their Newton polygons. Let $$\tilde W$$ be the Iwahori–Weyl group of $$G$$ over $$L$$. Let $$\{\mu\}$$ be a geometric conjugacy class of cocharacters of $$G$$. Let $$\text{Adm}(\{\mu\}) \subseteq \tilde W$$ be the admissible subset of $$\tilde W$$ ([13] and [16]) and $$B(G, \{\mu\})$$ be the finite subset of $$B(G)$$ defined by the group-theoretic version of Mazur’s theorem [12, Section 6]. The main result of this article is as follows. Theorem 1.1. The set $$B(G, \{\mu\})$$ contains a unique maximal element and this element is represented by an element in $$\text{Adm}(\{\mu\})$$. □ For quasi-split groups, this is obvious as the unique maximal element of $$B(G, \{\mu\})$$ is represented by a translation element. However, it is much more complicated for non quasi-split groups. This result is an important ingredient in the proof [8] of the Kottwitz–Rapoport conjecture [14, Conjecture 3.1] and [16, Conjecture 5.2] on the union of affine Deligne–Lusztig varieties. The knowledge of the explicit description of the maximal element of $$B(G, \{\mu\})$$ is also useful in the study of the $$\mu$$-ordinary locus, the most general Newton stratum, of Shimura varieties. In fact, Theorem 1.1 is a statement on the Iwahori–Weyl group $$\tilde W$$, the automorphism on $$\tilde W$$ induced from the Frobenius morphism $$\sigma$$ of $$G$$ and a cocharacter $$\mu$$ in $$\{\mu\}$$. In Section 2, we introduce a set $$B(\tilde W, \mu, \sigma)$$ (for any diagram automorphism $$\sigma$$ on $$\tilde W$$) and reformulate Theorem 1.1 as a statement on the triple $$(\tilde W, \mu, \sigma)$$. The relation between $$B(G, \{\mu\})$$ and $$B(\tilde W, \mu, \sigma)$$ is discussed in the Appendix. This reformulation allows us to relate different diagram automorphisms of the Iwahori–Weyl groups, which plays an essential role in the proof. We show that the set $$B(\tilde W, \mu, \sigma)$$ contains a unique maximal element in Section 3. It requires more work to show that this maximal element is represented by an element in the admissible set. We show in Section 4 that it suffices to consider the superbasic case and in Section 5 that it suffices to consider the irreducible case. In Section 6, we prove the statement for the irreducible superbasic case (that is, $$\sigma$$ is a diagram automorphism of order $$n$$ for the Iwahori–Weyl group of $$PGL_n$$). This completes the proof of Theorem 1.1. 2 Preliminaries 2.1 Let $$\mathfrak R=(X^*, R, X_*, R^\vee, \Pi)$$ be a based reduced root datum, where $$R \subseteq X^*$$ is the set of roots, $$R^\vee \subseteq X_*$$ is the set of coroots and $$\Pi \subseteq R$$ is the set of simple roots. Let $$\langle ~ , ~ \rangle : X^* \times X_* \rightarrow \mathbb Z$$ be the natural perfect pairing between $$X^*$$ and $$X_*$$. Let $$V=X_* \otimes \mathbb R$$. For any $$\alpha \in R$$, we have a reflection $$s_\alpha$$ on $$V$$ sending $$v$$ to $$v-\langle \alpha, v\rangle \alpha^\vee$$. The reflections $$s_\alpha$$ generate the finite Weyl group$$W_0$$ of $$R$$. Let $$\mathbb S=\{s_\alpha; \alpha \in \Pi\}$$ be the set of simple reflections. Then $$(W_0, \mathbb S)$$ is a Coxeter system. For any $$J \subseteq \mathbb S$$, let $$W_J$$ be the subgroup of $$W_0$$ generated by $$J$$ and $${}^J W_0=\{w \in W_0;$$$$w=\min(W_J w)\}$$. The closed dominant chamber is the set   C={v∈V;⟨α,v⟩≥0 for every α∈Π}. Then for any $$v \in V$$, the set $$\{w(v); w \in W_0\}$$ contains a unique element in $$\mathfrak C$$. We denote this element by $$\bar v$$. 2.2 Set $$\tilde R=R \times \mathbb Z$$. For $$(\alpha, k) \in \tilde R$$, we have an affine root $$\tilde \alpha=\alpha+k$$ and an affine reflection $$s_{\tilde \alpha}$$ on $$V$$ sending $$v$$ to $$v-(\langle\alpha, v\rangle - k) \alpha^\vee$$. For any affine root $$\tilde \alpha$$, let $$H_{\tilde \alpha}$$ be the hyperplane in $$V$$ fixed by the reflection $$s_{\tilde \alpha}$$. Set   Wa=ZR∨⋊W0={tλw;λ∈ZR∨,w∈W0},W~=X∗⋊W0={tλw;λ∈X∗,w∈W0}. We call $$W_a$$ the affine Weyl group and $$\tilde W$$ the extended affine Weyl group. Let $$\text{Aff}(V)$$ be the group of affine transformations on $$V$$. We realize both $$W_a$$ and $$\tilde W$$ as subgroups of $$\text{Aff}(V)$$, where $$t^\lambda$$ acts by translation $$v \mapsto v+\lambda$$ on $$V$$. We may identify $$W_a$$ with the subgroup of $$\text{Aff}(V)$$ generated by the affine reflections. Let $$R^+ \subseteq R$$ be the set of positive roots determined by $$\Pi$$. The base alcove is the set   a={v∈V;0<⟨α,v⟩<1 for every α∈R+}. The set of positive affine roots is $$\{\alpha+k; \alpha \in R^+, k \geq 1\} \cup \{-\alpha+k; \alpha \in R^+, k \geq 0\}$$. The affine simple roots are $$-\alpha$$ for $$\alpha \in \Pi$$ and $$\beta+1$$, where $$\beta$$ runs over maximal positive roots in $$R$$. Note that the positive roots in $$R$$ are not positive as affine roots. The hyperplanes $$H_{\tilde \alpha}$$, for the affine simple roots $$\tilde \alpha$$, are exactly the walls of the base alcove $$\mathbf a$$. Let $$\widetilde{\mathbb{S}}$$ be the set of $${{s}_{{\tilde{\alpha }}}}$$, where $$\tilde \alpha$$ runs over affine simple roots. Then $$\mathbb{S}\subseteq \widetilde{\mathbb{S}}$$ and $$({{W}_{a}},\widetilde{\mathbb{S}})$$ is a Coxeter group. Let $$\Omega$$ be the isotropy group in $$\tilde W$$ of the base alcove $$\mathbf a$$. Then $$\tilde W=W_a \rtimes \Omega$$. We extend the Bruhat order on $$W_a$$ to $$\tilde W$$ as follows: for $$w, w' \in W_a$$ and $$\tau, \tau' \in \Omega$$, we say that $$w \tau \le w' \tau'$$ if $$\tau=\tau'$$ and $$w \le w'$$ (with respect to the Bruhat order on the Coxeter group $$W_a$$). We put $$\ell(w \tau)=\ell(w)$$. 2.3 Let $$\sigma \in \text{Aff}(V)$$ be an automorphism of finite order such that $$\sigma(\mathbf a)=\mathbf a$$ and the conjugation action of $$\sigma$$ stabilizes $$\tilde W$$. Then $$\sigma$$ induces a bijection on the set of walls of $$\mathbf{a}$$ and hence a bijection on $$\widetilde{\mathbb{S}}$$. Let $$\varsigma \in GL(V)$$ denote the linear part of $$\sigma$$ with respect to the decomposition $$\text{Aff}(V)=V \rtimes GL(V)$$. Then $$\sigma$$ acts by conjugation on the translation subgroup of $$\text{Aff}(V)$$ via $$\varsigma$$:   Ad(σ)(tξ)=tς(ξ) for ξ∈V. Since we assume that $$\sigma$$ normalizes $$\tilde W$$, it follows that $$\varsigma$$ stabilizes $$X_*$$ and normalizes $$W_0$$. The $$\sigma$$-conjugation action on $$\tilde W$$ is defined by $$w \cdot_\sigma w'=w w' \text{Ad}(\sigma)(w)^{-1}$$. We have two invariants on the $$\sigma$$-conjugacy classes. Since $$\sigma(\mathbf a)=\mathbf a$$, the conjugation action of $$\sigma$$ stabilizes $$\Omega$$. Let $$\Omega_\sigma$$ be the set of $$\sigma$$-coinvariants on $$\Omega$$. The Kottwitz map $$\kappa_{\tilde W, \sigma}: \tilde W \to \Omega_\sigma$$ is obtained by composing the natural projection map $$\tilde W \to \tilde W/W_a \cong \Omega$$ with the projection map $$\Omega \to \Omega_\sigma$$. It is constant on each $$\sigma$$-conjugacy class of $$\tilde W$$. This gives one invariant. Another invariant is given by the Newton map. For any $$w \in \tilde W$$, we consider the element $$w \sigma \in \text{Aff}(V)$$. There exists $$n \in \mathbb N$$ such that $$(w \sigma)^n=t^\xi$$ for some $$\xi \in X_*$$. Let $$\nu_{w, \sigma}=\xi/n$$ and $$\bar \nu_{w, \sigma}$$ be the unique dominant element in the $$W_0$$-orbit of $$\nu_{w, \sigma}$$. We call $$\bar \nu_{w, \sigma}$$ the Newton point of $$w$$ (with respect to the $$\sigma$$-conjugation action). It is known that $$\nu_{w, \sigma}$$ is independent of the choice of $$n$$ and $$\bar \nu_{w, \sigma}=\bar \nu_{w', \sigma}$$ if $$w$$ and $$w'$$ are $$\sigma$$-conjugate. Moreover, $$w \sigma t^\xi=t^\xi w \sigma \in \text{Aff}(V)$$. Hence $$w \text{Ad}(\sigma)(t^{\xi})=t^\xi w$$. Therefore $$\varsigma(\xi)$$ and $$\xi$$ are in the same $$W_0$$-orbit and   ν¯w,σ=ς(νw,σ)¯. (a) 2.4 Let $$\mu$$ be a dominant cocharacter, that is $$\mu \in X_* \cap \mathfrak C$$. The $$\mu$$-admissible set is defined as   Adm(μ)={w∈W~;w≤tx(μ) for some x∈W0}. The partial order on $$\mathfrak C$$ is defined as follows. Let $$v, v' \in \mathfrak C$$. We say that $$v \leq v'$$ if $$v'-v \in \sum_{\alpha \in \Pi} \mathbb R_{\geq 0} \alpha^\vee$$. Let $$N$$ be the order of $$\sigma$$. For $$\mu \in X_*$$, we define   μσ♢=1N∑i=0N−1ςi(μ)¯∈Cμσ♣=νtμ,σ=1N∑i=0N−1ςi(μ). If $$\sigma(0)=0$$, then $$\mu^\diamondsuit_\sigma=\mu^\clubsuit_\sigma$$. Set   B(W~,μ,σ)={ν¯w,σ;w∈W~,κW~,σ(w)=κW~,σ(tμ),ν¯w,σ≤μσ♢}. The elements in $$B(\tilde W, \mu, \sigma)$$ are called the acceptable elements for $$\mu$$. The main result of this article is as follows. Theorem 2.1. (1) The set $$B(\tilde W, \mu, \sigma)$$ contains a unique maximal element $$\nu$$ (with respect to the partial order $$\leq$$ on $$\mathfrak C$$). (2) There exists an element $$w \in \text{Adm}(\mu)$$ with $$\bar \nu_{w, \sigma}=\nu$$. □ The relation between the sets $$B(\tilde W, \mu, \sigma)$$ and $$B(G, \{\mu\})$$ will be discussed in the Appendix. 3 The Maximal Element in $$B(\tilde W, \mu, \sigma)$$ 3.1 Let $$\mathfrak R_{ad}=(\mathbb Z R, R, X_*^{ad}, R^\vee, \Pi)$$ be the root datum of the adjoint group of the reductive group with root datum $$\mathfrak R$$. Here $$X_*^{ad}$$ is the dual lattice of $$\mathbb Z R$$. The perfect pairing $$\langle ~ , ~ \rangle : X^* \times X_* \rightarrow \mathbb Z$$ induces a natural map $$\pi: X_* \otimes \mathbb R \to X_*^{ad} \otimes \mathbb R$$. Set $$\tilde W_{ad}=X_*^{ad} \rtimes W_0$$. Then the map $$\pi$$ induces a natural map $$\pi: \tilde W \to \tilde W_{ad}$$. Lemma 3.1. Let $$v \in X_*^{ad} \otimes \mathbb R$$ and $$\hat{v}$$, $$\hat{v}'$$ be two lifts of $$v$$ under $$\pi$$. Then $$\pi \circ \sigma(\hat{v})=\pi \circ \sigma(\hat{v}')$$. □ Proof Note that $$\sigma(\hat{v})=\sigma(\hat{v}')+\sigma(\hat{v}-\hat{v}')-\sigma(0)$$. Since $$\varsigma$$ normalizes $$W_0$$, it gives a permutation of the hyperplanes $$H_\alpha=\{v \in V; \langle\alpha, v\rangle=0\}$$ for $$\alpha \in R$$. As $$\hat{v}-\hat{v}'$$ lies in the intersection of these hyperplanes, we have $$\sigma(\hat{v}-\hat{v}')-\sigma(0)=\varsigma(\hat{v}-\hat{v}')$$ is still in this intersection. In other words, $$\pi(\sigma(\hat{v}-\hat{v}')-\sigma(0))=0$$. Thus $$\pi \circ \sigma(\hat{v})=\pi \circ \sigma(\hat{v}')$$. ■ Now we define a map $$\sigma_{ad}: X_*^{ad} \otimes \mathbb R \to X_*^{ad} \otimes \mathbb R$$ by $$v \mapsto \pi \circ \sigma(\hat{v})$$, where $$\hat{v} \in V$$ is a lift of $$v \in X_*^{ad} \otimes \mathbb R$$ under $$\pi$$. By Lemma 3.1, $$\sigma_{ad}$$ is well defined. The affine transformation $$\sigma_{ad}$$ on $$X_*^{ad} \otimes \mathbb R$$ induces a conjugation action on $$\tilde W_{ad}$$. It is easy to see that $$\pi(\nu_{w, \sigma})=\nu_{\pi(w), \sigma_{ad}}$$ for $$w \in \tilde W$$ and $$\pi$$ induces a bijection of posets from $$B(\tilde W, \mu, \sigma)$$ to $$B(\tilde W_{ad}, \pi(\mu), \sigma_{ad})$$. The map $$\pi$$ also induces a bijection of posets from $$\text{Adm}(\mu)$$ to $$\text{Adm}(\pi(\mu))$$. Thus Theorem 2.1 holds for $$B(\tilde W, \mu, \sigma)$$ if and only if it holds for $$B(\tilde W_{ad}, \pi(\mu), \sigma_{ad})$$. Lemma 3.2. If $$\mathfrak R=\mathfrak R_{ad}$$, then $$\Omega$$ acts simply transitively on the set of special vertices of $$\mathbf a$$. □ Remark 3.3. This Lemma is known to experts. We include a proof for the reader’s convenience. □ Proof First, the action of $$\Omega$$ on $$V$$ stabilizes the set of special vertices of $$\mathbf a$$. Let $$v \in V$$ be a special vertex of $$\mathbf a$$. By [1, VI Section 2 Prop. 3], $$v \in X_*$$. Since $$W_0$$ acts simply transitively on the set of chambers, there exists a unique $$x \in W_0$$ such that $$x t^{-v}$$ sends $$\mathbf a$$ to an alcove $$\mathbf a'$$ in the dominant chamber. Since $$x t^{-v}(v)=0$$, we deduce that $$0$$ lies in the closure of $$\mathbf a'$$. Note that $$\mathbf a$$ is the unique alcove in the dominant chamber whose closure contains $$0$$. Hence $$\mathbf a'=\mathbf a$$ and $$x t^{-v} \in \Omega$$. In other words, there exists an element in $$\Omega$$ sending $$v$$ to $$0$$. So $$\Omega$$ acts transitively on the set of special vertices of $$\mathbf a$$. Let $$w \in \tilde W$$. If $$w$$ preserves $$0$$, then $$w \in W_0$$. Note that $$\Omega \cap W_0=\{1\}$$. Thus the action of $$\Omega$$ on the set of special vertices is simply transitive. ■ 3.2 In the rest of this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$. By Lemma 3.2, there exists $$\tau \in \Omega$$ such that $$\sigma_0:=\tau^{-1} \sigma$$ preserves $$0 \in V$$. In the rest of the article, unless otherwise stated, we denote by $$\lambda$$ the dominant cocharacter with $$\tau \in t^\lambda W_0$$. Notice that $$\sigma_0$$ is a linear action on $$V$$ and the conjugation action of $$\sigma_0$$ stabilizes the subset $$\mathbb S$$ of $$\tilde{\mathbb S}$$. For simplicity, in the rest of the article, we will say $$\sigma_0$$-orbits in $$\mathbb S$$ instead of $$\text{Ad}(\sigma_0)$$-orbits in $$\mathbb S$$. %Then $$\sigma_0$$ stabilizes $$\mathbb S$$ inside $$\tilde{\mathbb S}$$ and is a diagram automorphism of $$W_0$$. In particular, $$\sigma_0$$ is a linear action on $$V$$. By definition, $$\nu_{w, \sigma}=\nu_{w \tau, \sigma_0}$$ for all $$w \in \tilde W$$. By 2.3 (a),   ν¯w,σ=ν¯wτ,σ0∈Cσ0:={v∈C;σ0(v)=v}. Lemma 3.4. If $$\xi \in X_* \cap \mathfrak C$$, then $$\xi^\diamondsuit_\sigma=\xi^\clubsuit_{\sigma_0}=\nu_{t^\xi, \sigma_0}$$. □ Proof Note that $$\sigma_0=\tau^{-1} \sigma$$ and is linear. We have $$\sigma_0=x \varsigma$$ for some $$x \in W_0$$. Since $$\varsigma$$ normalizes $$W_0$$, we have $$\sigma_0^i \in W_0 \xi^i$$ for any $$i \in \mathbb Z$$. Therefore $$\overline{\varsigma^i(\zeta)}=\overline{\sigma_0^i(\zeta)}$$. Since $$\sigma_0$$ stabilizes $$\mathfrak C$$, $$\xi^\diamondsuit_\sigma=\xi^\diamondsuit_{\sigma_0}=\xi^\clubsuit_{\sigma_0}$$. ■ 3.3 For any $$i \in \mathbb S$$, let $$\omega^\vee_i \in V$$ be the corresponding fundamental coweight and $$\alpha_i^\vee \in V$$ be the corresponding simple coroot. We denote by $$\omega_i, \alpha_i \in V^*$$ the corresponding fundamental weight and corresponding simple root, respectively. For each $$\sigma_0$$-orbit $$c$$ of $$\mathbb S$$, we set   ωc=∑i∈cωi. For any $$v \in \mathfrak C$$, we set   J(v)={s∈S;s(v)=v},I(v)=S∖J(v). If $$v=\sigma_0(v)$$, then both $$J(v)$$ and $$I(v)$$ are $$\sigma_0$$-stable. The following lemma is essentially contained in [2, Section 7.1]. Due to its importance, we provide a proof for completeness. Lemma 3.5. Let $$v \in \mathfrak C^{\sigma_0}$$. Then $$v =\nu_{w, \sigma}$$ for some $$w \in t^\mu W_a$$ if and only if $$\langle\omega_c, \mu^\clubsuit_{\sigma_0} + \lambda^\clubsuit_{\sigma_0}-v\rangle \in\mathbb Z$$ for any $$\sigma_0$$-orbit $$c$$ of $$I(v)$$. □ Proof Suppose $$\nu_{w, \sigma}=v$$. We have $$w \tau=t^\gamma x$$ for some $$\gamma \in X_*$$ and $$x \in W_0$$. By definition, $$(w\sigma)^n=t^{n v}$$ for some $$n \in \mathbb N$$. Since $$t^{n v}$$ commutes with $$w\sigma=t^\gamma x\sigma_0$$, we have $$x\sigma_0(v)=x(v)=v$$. Thus $$x \in W_{J(v)}$$. Let $$N_0$$ be the order of the finite subgroup of $$\text{Aff}(V)$$ generated by $$W_0$$ and $$\sigma_0$$. Then   νw,σ=νwτ,σ0=1N0∑k=0N0−1(xσ0)k(γ)=1N0∑k=0N0−1(xAd(σ0)(x)⋯Ad(σ0)k−1(x))σ0k(γ)∈1N0∑k=0N0−1σ0k(γ)+∑j∈J(v)Qαj∨=γσ0♣+∑j∈J(v)Qαj∨. If $$w \in t^\mu W_a$$, then $$w \tau \in t^{\mu+\lambda} W_a$$ and $$\mu+\lambda-\gamma \in \mathbb Z R^\vee$$. Hence   ⟨ωc,μσ0♣+λσ0♣−v⟩=⟨ωc,μσ0♣+λσ0♣−γσ0♣⟩=⟨ωc,μ+λ−γ⟩∈Z. On the other hand, suppose $$a_c=\langle\omega_c, \mu^\clubsuit_{\sigma_0} + \lambda^\clubsuit_{\sigma_0}-v\rangle \in \mathbb Z$$ for each $$\sigma_0$$-orbit $$c$$ of $$I(v)$$. We also set $$a_c=0$$ if $$c \nsubseteq I(v)$$. We construct an element $$w \in t^\mu W_a$$ such that $$\nu_{w,\sigma}=v$$. For each $$\sigma_0$$-orbit of $$J(v)$$, we choose a representative. Let $$y$$ be the product of these representatives (in some order). Then $$y$$ is a $$\sigma_0$$-twisted Coxeter element of $$W_{J(v)}$$ in the sense of [18, 7.3]. For each $$\sigma_0$$-orbit $$c$$ of $$I(v)$$, we choose a representative $$i_c$$. Let $$\alpha^\vee_{i_c}$$ be the corresponding simple coroot. Set $$\beta=\mu+\lambda-\sum_c a_c \alpha^\vee_{i_c}$$ and $$w=t^{\beta} y \tau^{-1} \in t^\mu W_a$$. Write $$\beta=h+r$$ with $$r \in \sum_{j \in J(v)} \mathbb Q \alpha_j^\vee$$ and $$h \in \sum_{i \in I(v)} \mathbb Q \omega_i^\vee$$. Then   νw,σ=νwτ,σ0=1N0∑k=0N0−1(yσ0)k(β)=hσ0♣+1N0∑k=0N0−1(yσ0)k(r)=hσ0♣=μσ0♣+λσ0♣−∑cac(αic∨)σ0♣−rσ0♣, where the fourth equality follows from [18, Lemma 7.4]. Hence for any $$\sigma_0$$-orbit $$c$$ of $$I(v)$$ and any $$j \in J(v)$$, we have   ⟨ωc,μσ0♣+λσ0♣−νw,σ⟩=⟨ωc,∑c′ac′(αic′∨)σ0♣⟩=ac and   ⟨αj,μσ0♣+λσ0♣−νw,σ⟩=⟨αj,μσ0♣+λσ0♣⟩=⟨αj,μσ0♣+λσ0♣−v⟩, which means $$\nu_{w,\sigma}=v$$ as desired. ■ Corollary 3.6. $$\mu^\diamondsuit_{\sigma}=\mu^\clubsuit_{\sigma_0} \in B(\tilde W, \mu, \sigma)$$ if and only if $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$ for any $$\sigma_0$$-orbit $$c$$ of $$I(\mu^\diamondsuit_{\sigma})$$. In this case, $$\mu^\diamondsuit_{\sigma}$$ is a priori the unique maximal element of $$B(\tilde W, \mu, \sigma)$$. □ 3.4 We follow [2, Section 6]. For any $$\sigma_0$$-stable subset $$B$$ of $$\mathfrak C$$, we define   C≥B={v∈Cσ0;v≥b,∀b∈B}. We say $$B$$ is reduced if $$C_{\geq B} \subsetneq C_{\geq B'}$$ for any $$\sigma_0$$-stable proper subset $$B' \subsetneq B$$. For any $$i \in \mathbb S$$, let   pr(i):V=Rωi∨⊕∑j≠iRαj∨→Rωi∨ be the projection map. Now we prove part (1) of Theorem 2.1. 3.5 Proof of Theorem 2.1 (1) By Section 3.1, it suffices to consider the case where $$\mathfrak R=\mathfrak R_{ad}$$. For any $$i \in \mathbb S$$, let $$c$$ denote the $$\sigma_0$$-orbit of $$i$$, and define $$e_i \in \mathbb Q \omega_i^\vee$$ by   ⟨ωi,ei⟩=1#cmax({t∈⟨ωc,μσ0♣+λσ0♣⟩+Z;t≤⟨ωc,μσ0♣⟩}∪{0}). Let $$E_0=\{e_i; i \in \mathbb S\}$$. It is easy to prove by induction on the number of $$\sigma_0$$-orbits on $$E_0$$ that there exists a $$\sigma_0$$-stable subset $$E$$ of $$E_0$$ which is reduced and satisfies $$C_{\geq E}=C_{\geq E_0}$$. Let $$I(E)=\{i \in \mathbb S; e_i \in E\}$$. By [2, Theorem 6.5] In fact, we use here a “$$\sigma_0$$-fixed” version of [2, Theorem 6.5], which can be proved in the same way as in loc.cit., there exists an element $$\nu \in C_{\geq E}$$ defined by $$I(\nu)=I(E)$$ and $$\langle\omega_j, \nu\rangle=\langle\omega_j, e_j\rangle$$ for $$j \in I(E)$$, which satisfies $$C_{\geq \nu}=C_{\geq E}=C_{\geq E_0}$$. Since $$\mu^\diamondsuit_\sigma=\mu^\clubsuit_{\sigma_0} \in C_{\geq E_0}$$, we have $$\nu \leq \mu^\diamondsuit_\sigma$$. By Lemma 3.5, $$\nu \in B(\tilde W, \mu, \sigma)$$. (In fact, we use here a “$$\sigma_0$$-fixed” version of [2, Theorem 6.5], which can be proved in the same way as in loc.cit.) Since $$\nu \in C_{\geq E_0}$$, $$\nu \geq e_i$$ for any $$i \in \mathbb S$$. Therefore, for any $$\sigma_0$$-orbit $$c$$ of $$\mathbb S$$, we have   ⟨ωc,μσ0♣⟩≥⟨ωc,ν⟩≥∑j∈c⟨ωj,ej⟩≥⟨ωc,μσ0♣+λσ0♣⟩−⌈⟨ωc,λσ0♣⟩⌉, (a) where the last inequality follows from our definition of $$e_j$$ for $$j \in \mathbb S$$. Let $$\nu' \in B(\tilde W,\mu, \sigma)$$. Set $$E(\nu')=\{pr_{(j)}(\nu'); j \in I(\nu')\}$$. By Lemma 3.5 and the inequality $$\nu' \leq \mu^\diamondsuit_\sigma=\mu^\clubsuit_{\sigma_0}$$, we have, for any $$\sigma_0$$-orbit $$c$$ of $$I(\nu')$$ and $$j \in c$$, that   #c⋅⟨ωj,pr(j)(ν′)⟩=#c⋅⟨ωj,ν′⟩=⟨ωc,ν′⟩∈⟨ωc,μσ0♣+λσ0♣⟩+Z and   #c⋅⟨ωj,pr(j)(ν′)⟩≤#c⋅⟨ωj,μσ0♣⟩=⟨ωc,μσ0♣⟩. So $$\langle\omega_j, pr_{(j)}(\nu')\rangle \leq \langle\omega_j, e_j\rangle$$, that is, $$pr_{(j)}(\nu') \leq e_j \leq \nu$$ for $$j \in I(\nu')$$. By [2, Lemma 6.2 (i)], we deduce that $$\nu' \leq \nu$$. Therefore $$\nu$$ is the unique maximal element of $$B(\tilde W,\mu, \sigma)$$. 4 Reduction to the Superbasic Case 4.1 In the rest of the article, we prove Theorem 2.1 (2), beginning in this section with a reduction step to the superbasic case. For any element $$w \sigma^i$$ with $$w \in \tilde W$$ and $$i \in \mathbb Z$$, we put $$\ell(w \sigma^i)=\ell(w)$$. This is well-defined since $$\sigma(\mathbf a)=\mathbf a$$. Let $$\epsilon=w \sigma^i$$ with $$\ell(\epsilon)=0$$. Then the conjugation action of $$\epsilon$$ on $$\tilde W$$ sends simple reflections to simple reflections. We say that $$\epsilon$$ is superbasic (for $$\tilde W$$) if each $$\text{Ad}(\epsilon)$$-orbit on $$\tilde{\mathbb S}$$ is a union of connected components of the affine Dynkin diagram of $$\tilde W$$. By [9, 3.5], $$\epsilon$$ is superbasic if and only if $$W_a=W_1^{m_1} \times \cdots \times W_l^{m_l}$$, where each $$W_i$$ is an extended affine Weyl group of type $$\tilde A_{n_i-1}$$ and $$\epsilon$$ gives an order $$n_i m_i$$ permutation on the set of simple reflections of $$W_i^{m_i}$$. 4.2 Let $$J \subseteq \mathbb S$$. Let $$\tilde W_J=X_* \rtimes W_J$$ be the corresponding parabolic subgroup of $$\tilde W$$. This is the extended affine Weyl group associated to the root datum $$\mathfrak R_J=(X^*, R_J, X_*, R^\vee_J, \Pi_J)$$, where $$\Pi_J$$ is the subset of simple roots corresponding to $$J$$ and $$R_J \subseteq R$$ is the set of roots spanned by $$\Pi_J$$. Set $$R^+_J=R_J \cap R^+$$. Let $$\mathbf a_J=\{v \in V; 0 < \langle \alpha, v\rangle < 1 \text{ for every } \alpha \in R^+_J\}$$ be the base alcove associated to $$\tilde W_J$$. The set of positive affine roots $$\tilde R_J$$ for $$\tilde W_J$$ is $$\{\alpha+k; \alpha \in R^+_J, k \geq 1\} \cup \{-\alpha+k; \alpha \in R^+_J, k \geq 0\}$$. The affine simple roots for $$\tilde W_J$$ are $$-\alpha$$ for $$\alpha \in \Pi_J$$ and $$\beta+1$$, where $$\beta$$ runs over maximal positive roots in $$R_J^+$$. Note that the positive roots in $$R_J$$ are not positive as affine roots in $$\tilde R_J$$. We denote by $$\le_J$$ and $$\ell_J$$ the Bruhat order and length function on $$\tilde W_J$$. Although $$\tilde W_J$$ is a subgroup of $$\tilde W$$, $$\le_J$$ and $$\ell_J$$ can be quite different from the restrictions of $$\le$$ and $$\ell$$ to $$\tilde W_J$$. 4.3 In the rest of this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$. We take $$\tilde t \in \Omega$$ and $$\sigma_0=\tau^{-1} \sigma$$ as in § 3.2. Recall that $$\lambda$$ is the dominant cocharacter with $$\tau \in t^\lambda W_0$$. We will associate to $$\sigma$$ a superbasic element for a parabolic subgroup of $$\tilde W$$ and reduce Theorem 2.1 (2) to the superbasic case. We follow the approach in [10, Section 5]. Let $$V^\sigma$$ be the fixed point set of $$\sigma$$. Since $$\sigma$$ is an affine transformation on $$V$$ of finite order, $$V^\sigma$$ is a nonempty affine subspace. Set $$V'=\{v-e; v \in V^\sigma\}$$, where $$e$$ is an arbitrary point of $$V^\sigma$$. Then $$V'$$ is the (linear) subspace of $$V$$ parallel to $$V^\sigma$$. We choose a generic point $$v_0$$ of $$V'$$, that is for any root $$\alpha \in R$$, $$\langle\alpha, v_0\rangle=0$$ implies that $$\langle\alpha, v'\rangle=0$$ for all $$v' \in V'$$. We set $$I=I(\bar v_0)$$, $$J=J(\bar v_0)$$ and $$\sigma^J=z \sigma z^{-1} \in \tilde W \sigma$$, where $$z$$ is the unique element in $${}^J W_0$$ with $$\bar v_0=z(v_0)$$. Lemma 4.1. (1) The set $$J$$ is stable under $$\sigma_0$$-conjugation. (2) $$z(\lambda)^\clubsuit_{\sigma_0} \in \mathbb Q R_J^\vee$$. (3) The element $$\sigma^J$$ is a superbasic element for $$\tilde W_J$$. □ Proof (1) By definition, $$\sigma(0)=\lambda$$. Hence $$\sigma(v_0)=v_0+\lambda$$ and $$\sigma^J(\bar v_0)=\bar v_0+z(\lambda)$$. Write $$\sigma^J$$ as $$\sigma^J=t^{z(\lambda)}u \sigma_0$$ for some $$u \in W_0$$. Then $$u\sigma_0(\bar v_0)=\bar v_0$$. Therefore $$\sigma_0(\bar v_0)=u^{-1}(\bar v_0)$$ is the unique dominant element in the $$W_0$$-orbit of $$v_0$$. Hence $$\bar v_0=\sigma_0(\bar v_0)=u^{-1}(\bar v_0)$$. Therefore $$u^{-1}n W_J$$ and $$\text{Ad}(\sigma_0)(J)=J$$. (2) Since $$\sigma^J$$ is of finite order, we have $$(\sigma^J)^m=1$$ for some $$m \in \mathbb N$$. On the other hand, using the expression $$\sigma^J=t^{z(\lambda)}u \sigma_0$$, one computes that $$(\sigma^J)^m=t^{\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda))}=1$$. So $$\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda))=0$$. Since $$u^{-1}n W_J$$ and $$\sigma_0(R_J^\vee)=R_J^\vee$$, we have $$(u\sigma_0)^k(z(\lambda))-\sigma_0^k(z(\lambda)) \in \mathbb Z R_J^\vee$$ for $$k \in \mathbb Z$$. Thus $$(z(\lambda))^\clubsuit_{\sigma_0} \in \frac{1}{m}\sum_{k=0}^{m-1} (u\sigma_0)^k(z(\lambda)) + \mathbb Q R_J^\vee=\mathbb Q R_J^\vee$$. (3) Since $$\sigma_0$$ stabilizes $$\mathbf a_J$$, the length function $$\ell_J$$ on $$\tilde W_J$$ extends to the subgroup of $$\text{Aff}(V)$$ generated by $$\tilde W_J$$ and $$\sigma_0$$ via the usual rule $$\ell_J(w \sigma_0^i)=\ell_J(w)$$ for $$w \in \tilde W_J$$ and $$i \in \mathbb Z$$. Since $$z^{-1}(R_J^+) \subseteq R^+$$, we have $$z(\mathbf a) \subseteq \mathbf a_J$$. In other words, $$\mathbf a_J$$ is the unique alcove associated to $$\tilde W_J$$ that contains $$z(\mathbf a)$$. Since $$\sigma^J(z(\mathbf a))=z(\mathbf a)$$, $$\sigma^J(\mathbf a_J)$$ is also the unique alcove associated to $$\tilde W_J$$ that contains $$z(\mathbf a)$$. Therefore $$\sigma^J(\mathbf a_J)=\mathbf a_J$$. Since $$v_0$$ is generic in $$V'$$, $$\bar v_0=z(v_0)$$ is generic in $$z(V')$$. So each point of $$z(V')$$ is fixed by $$W_J$$. Therefore, for any $$\tilde \alpha \in \tilde R_J$$, either $$V^{\sigma^J} \cap H_{\tilde \alpha} = \emptyset$$ or $$V^{\sigma^J} \subseteq H_{\tilde \alpha}$$, where $$V^{\sigma^J}=z(V^\sigma)$$ is the fixed-point set of $$\sigma^J$$ on $$V$$. Since $$\sigma^J(\mathbf a_J)=\mathbf a_J$$ and $$\sigma^J$$ is of finite order, $$\mathbf a_J$$ contains a fixed point of $$\sigma^J$$. So $$V^{\sigma^J} \nsubseteq H_{\tilde \alpha}$$ and hence $$V^{\sigma^J} \cap H_{\tilde \alpha}=\emptyset$$. By [9, Proposition 3.5] (for $$\tilde W_G:=\tilde W_J, J_\mathcal O:=J$$ and $$y=1$$), $$\sigma^J$$ is superbasic for $$\tilde W_J$$. ■ Lemma 4.2. Let $$c$$ be a $$\sigma_0$$-orbit of $$\mathbb S$$. Then $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$ if $$c \subseteq I$$. □ Proof Write $$\lambda=z(\lambda)+\theta$$ for some $$\theta \in \mathbb Z R^\vee$$. We have   ⟨ωc,λσ0♣⟩=⟨ωc,z(λ)σ0♣⟩+⟨ωc,θ⟩≡⟨ωc,z(λ)σ0♣⟩modZ. By Lemma 4.1 (2), $$\langle\omega_c, z(\lambda)^\clubsuit_{\sigma_0}\rangle=0$$ if $$c \subseteq I$$. The proof is finished. ■ Proposition 4.3. The maximal Newton point of $$B(\tilde W, \mu, \sigma)$$ is contained in the natural inclusion $$B(\tilde W_J, \mu, \sigma^J) \hookrightarrow B(\tilde W, \mu, \sigma)$$. □ Proof For any $$j \in J$$, we denote by $$\omega_j^J$$ the fundamental weight corresponding to $$j$$ in the root datum $$\mathfrak R_J$$. We set $$\omega_c^J=\sum_{j \in c} \omega_j^J$$ for any $$\sigma_0$$-orbit of $$c$$ of $$J$$. Let $$\nu$$ be the maximal Newton point of $$B(\tilde W, \mu, \sigma)$$. Let $$c$$ be a $$\sigma_0$$-orbit of $$I$$. By Lemma 4.2, $$\langle\omega_c, \lambda^\clubsuit_{\sigma_0}\rangle \in \mathbb Z$$. Applying Lemma 3.5 (a), we see that $$\langle\omega_c, \mu^\clubsuit_{\sigma_0}\rangle=\langle\omega_c, \nu\rangle$$ and $$\mu^\clubsuit_{\sigma_0} - \nu^{-1}n \mathbb Q R_J^\vee$$. By Lemma 4.1 (2), $$z(\lambda)^\clubsuit_{\sigma_0} \in \mathbb Q R_J^\vee$$. Thus $$\mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu^{-1}n \mathbb Q R_J^\vee$$. Now let $$c'$$ be a $$\sigma_0$$-orbit in $$I(\nu) \cap J$$. Then   ⟨ωc′J,μσ0♣+z(λ)σ0♣−ν⟩=⟨ωc′,μσ0♣+z(λ)σ0♣−ν⟩=⟨ωc′,μσ0♣+λσ0♣−ν⟩−⟨ωc′,θ⟩, where $$\theta=\lambda-z(\lambda) \in \mathbb Z R^\vee$$. By Lemma 3.5 $$\langle\omega_{c'}, \mu^\clubsuit_{\sigma_0}+\lambda^\clubsuit_{\sigma_0}-\nu\rangle \in \mathbb Z$$. Hence $$\langle\omega_{c'}^J, \mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu\rangle \in \mathbb Z$$. Again by Lemma 3.5, we have $$\pi_J(\nu) \in B((\tilde W_J)_{ad}, \pi_J(\mu), (\sigma^J)_{ad})$$, where $$\pi_J$$ and (respectively $$(\sigma^J)_{ad}$$) is defined similarly as $$\pi$$ (respectively $$\sigma_{ad}$$) in § 3.1 with $$\mathfrak R$$ and $$\sigma$$ replaced by $$\mathfrak R_J$$ and $$\sigma^J$$ , respectively. Since $$\mu^\clubsuit_{\sigma_0}+z(\lambda)^\clubsuit_{\sigma_0}-\nu^{-1}n \mathbb Q R_J^\vee$$ and   πJ:B(W~J,μ,σJ)→B((W~J)ad,πJ(μ),(σJ)ad) is a bijection of posets, we deduce that $$\nu^{-1}n B(\tilde W_J, \mu, \sigma^J)$$ as desired. ■ Lemma 4.4. Let $$K \subseteq \mathbb S$$ and $$z \in {}^K W_0$$. If $$w, w' \in \tilde W_K$$ with $$w \le_K w'$$ for the Bruhat order of $$\tilde W_K$$, then $$z^{-1} w z \le z^{-1} w' z$$ for the Bruhat order of $$\tilde W$$. □ Proof By the definition of Bruhat order, there exist positive affine roots $$\tilde \alpha_1, \cdots, \tilde \alpha_k$$ of $$\tilde W_K$$ such that   w≤Kwsα~1≤Kwsα~1sα~2≤K⋯≤Kwsα~1⋯sα~k=w′. Hence for any $$i$$, $$w s_{\tilde \alpha_1} \cdots s_{\tilde \alpha_i}(\tilde \alpha_{i+1})$$ is a positive affine root of $$\tilde W_K$$. Notice that $$z^{-1}$$ sends positive affine roots of $$\tilde W_K$$ to positive affine roots of $$\tilde W$$. Set $$\tilde \beta_i=z^{-1}(\tilde \alpha_i)$$. This is a positive affine root of $$\tilde W$$. Moreover, $$(z^{-1} w z) s_{\tilde \beta_1} \cdots s_{\tilde \beta_i}(\tilde \beta_{i+1})=z^{-1} w s_{\tilde \alpha_1} \cdots s_{\tilde \alpha_i} (\tilde \alpha_{i+1})$$ is a positive affine root of $$\tilde W$$. Thus   z−1wz≤z−1wzsβ~1≤z−1wzsβ~1sβ~2≤⋯≤z−1wzsβ~1⋯sβ~k=z−1w′z. ■ Corollary 4.5. If Theorem 2.1 (2) holds for $$B(\tilde W_J, \mu, \sigma^J)$$, then it holds for $$B(\tilde W, \mu, \sigma)$$. □ Proof Let $$\nu$$ be the maximal Newton point of $$B(\tilde W, \mu, \sigma)$$, which is also the maximal Newton point of $$B(\tilde W_J, \mu, \sigma^J)$$ by Proposition 4.3. By assumption, there exist $$w_1 \in t^\mu (W_a \cap \tilde W_J)$$ and $$x_1 \in W_J$$ such that $$\bar \nu_{w_1, \sigma^J}^J=\nu$$ and $$w_1 \le_J t^{x_1(\mu)}$$, where $$\bar \nu^J_{w_1, \sigma^J}$$ stands for the Newton point of $$w_1 \in \tilde W_J$$ defined with respect to the $$\sigma^J$$-conjugation action on $$\tilde W_J$$. Let $$z$$ be the element defined in Section 4.3. Let $$w=z^{-1} w_1 z$$ and $$x=z^{-1} x_1$$. Then we have $$\bar \nu_{w, \sigma}=\nu$$, $$w \in t^\mu W_a$$ and $$w \le t^{x(\mu)}$$ as desired. ■ 5 Reduction to the Irreducible Case 5.1 In this section, we assume that $$\mathfrak R=\mathfrak R_{ad}$$ and $$\sigma$$ acts transitively on the set of connected components of the affine Dynkin diagram of $$W_a$$. In other words, $$\tilde W=\tilde W_1 \times \cdots \times \tilde W_m$$, where $$\tilde W_1 \cong \cdots \cong \tilde W_m$$ are extended affine Weyl groups of adjoint type with connected affine Dynkin diagram and $$\text{Ad}(\sigma)(\tilde W_1)=\tilde W_2, \cdots, \text{Ad}(\sigma)(\tilde W_m)=\tilde W_1$$. Let $$W_i$$ be the finite Weyl group associated to $$\tilde W_i$$. As in Section 3.2, we write $$\sigma$$ as $$\sigma=\tau \sigma_0$$ with $$\tau \in \Omega$$ and $$\text{Ad}(\sigma_0)(\mathbb S)=\mathbb S$$. Write $$\mu$$ as $$\mu=(\mu_1, \cdots, \mu_m)$$, where each $$\mu_i$$ is a dominant cocharacter for $$\tilde W_i$$. Let $$y=(w_1, \dots, w_m) \in \tilde W$$. Then the $$m$$-th component of $$(y\sigma)^m \sigma^{-m} \in \tilde W$$ is $$w_m\cdots \text{Ad}(\sigma^{m-1})(w_1)$$. The map $$\bar \nu_{(w_1, \dots, w_m), \sigma} \mapsto \bar \nu_{w_m\cdots \text{Ad}(\sigma^{m-1})(w_1), \sigma^m}$$ induces a natural surjection from $$B(\tilde W, \mu, \sigma)$$ to $$B(\tilde W_m, \gamma, \sigma^m)$$, where $$\gamma=\sum_{i=1}^m \sigma_0^{m-i} (\mu_i)$$. Since the elements in $$B(\tilde W, \mu, \sigma)$$ are $$\sigma_0$$-invariant, it is in fact a bijection, whose inverse is given by $$v \mapsto \frac{1}{m}(\sigma_0(v), \sigma_0^2(v), \cdots, v)$$. It is easy to see this bijection is a bijection of posets. Lemma 5.1. If Theorem 2.1 (2) holds for $$(\tilde W_m, \gamma, \sigma^m)$$, then it holds for $$(\tilde W, \mu, \sigma)$$. □ Proof Let $$\nu$$ be the maximal element in $$B(\tilde W_m, \gamma, \sigma^m)$$. Then the maximal element in $$B(\tilde W, \mu, \sigma)$$ is $$\frac{1}{m}(\sigma_0(\nu), \cdots, \nu)$$. By assumption, there exists $$w \in \text{Adm}(\gamma)$$ such that $$\bar \nu_{w, \sigma^m}=\nu$$. By definition, there exists $$x \in W_m$$ such that $$w \le t^{x(\gamma)}$$. Since $$\ell(t^{x(\gamma)})=\sum_{i=1}^m \ell(t^{x(\sigma_0^{m-i}(\mu_i))})$$, there exists $$w_i \in \tilde W_m$$ for each $$i$$ such that $$w=w_m \cdots w_1$$ and $$w_i \le t^{x(\sigma_0^{m-i}(\mu_i))}$$ for all $$i$$. It is easy to see that $$\sigma^{i-m}=\tau'_i \sigma_0^{i-m}$$ for some $$\tau'_i \in \tilde W$$. Hence   Ad(σ)i−m(wi)≤Ad(σ)i−m(tx(σ0m−i(μi)))=Ad(τi′)Ad(σ0)i−m(tx(σ0m−i(μi)))=txi(μi) for some $$x_i \in W_i$$. Set $$y=(\text{Ad}(\sigma)^{1-m}(w_1), \cdots, w_m) \in \tilde W$$. Then $$y \in \text{Adm}(\mu)$$. Notice that the $$m$$-th component of $$(y \sigma)^m \sigma^{-m}$$ is $$w_m \cdots w_1=w$$. Hence $$\bar \nu_{y, \sigma}=\frac{1}{m}(\sigma_0(\nu), \cdots, \nu)$$. ■ 6 The Irreducible Superbasic Case 6.1 In this section, we consider the extended affine Weyl group $$\tilde W=\mathbb Z^n \rtimes \mathfrak{S}_n$$ of type $$\tilde A_{n-1}$$, where $$\mathfrak{S}_n$$ is the permutation group of $$\{1, 2, \dots, n\}$$ which acts on $$\mathbb Z^n \cong \oplus_{i=1}^n \mathbb Z e_i^\vee$$ by $$w(e_i^\vee)=e_{w(i)}^\vee$$ for $$w \in \mathfrak{S}_n$$. Let $$\{e_i\}_{i=1, \cdots, n}$$ be the dual basis. Set $$d=\sum_{i=1}^n e_i$$ and $$d^\vee=\sum_{i=1}^n e_i^\vee$$. The simple roots, fundamental weights, and fundamental coweights are given by $$\alpha_i=e_i-e_{i+1}$$, $$\omega_{i,n}=-\frac{i}{n}d+\sum_{j=1}^i e_j$$ and $$\omega_{i,n}^\vee=-\frac{i}{n}d^\vee+\sum_{j=1}^i e_j^\vee$$ , respectively for $$i$$ with $$1 \le i \le n-1$$. Then $$\tilde W_{ad}=(\oplus_{i=1}^{n-1} \mathbb Z \omega_{i, n}^\vee) \rtimes \mathfrak{S}_n$$, see Section 3.1. Set $$\varpi_{m,n}=\sum_{j=1}^m e_j^\vee$$. The map $$\pi$$ in Section 3.1 can be described explicitly as the $$\mathbb Q$$-linear projection $$\pi: \mathbb Q^n \to \mathbb Q R^\vee \subseteq \mathbb Q^n$$ such that $$d^\vee \mapsto 0$$ and $$\varpi_{m,n} \mapsto \omega_{m,n}^\vee$$ for $$m$$ with $$1 \le m \le n-1$$. We also denote the induced (surjective) projection $$\tilde W \to \tilde W_{ad}$$ by $$\pi$$. For any positive integer $$m<n$$, let $$\sigma_{m, n}=t^{\varpi_{m,n}} u_{m,n} \in t^{\varpi_{m,n}} \mathfrak{S}_n$$ be the unique length zero element with $$u_{m, n} \in \mathfrak{S}_n$$. Then any superbasic element in $$\tilde W_{ad}$$ is of the form $$\pi(\sigma_{m, n})$$ for some positive integer $$m<n$$ co-prime to $$n$$. The main purpose of this section is to prove the following result. proposition 6.1. Let $$m<n$$ be a positive integer co-prime to $$n$$. Let $$\mu \in \oplus_{i=1}^n \mathbb Z e_i^\vee$$ be a dominant cocharacter. Then there exist $$\tilde W \in \tilde W$$ and $$x \in \mathfrak{S}_n$$ such that $$\tilde W \le t^{x(\mu)}$$ and $$\pi(\bar \nu_{\tilde W, \sigma_{m,n}})=\bar \nu_{\pi(\tilde W), \pi(\sigma_{m, n})}$$ equals the unique maximal Newton point $$\nu$$ of $$B(\tilde W_{ad}, \pi(\mu), \pi(\sigma_{m,n}))$$. □ The proof will be given in Section 6.6. 6.2 We first show that Proposition 6.1 implies Theorem 2.1 (2) for any triple $$(\tilde W_1, \mu_1, \sigma_1)$$. Let $$\mathfrak R$$ be the root datum of $$\tilde W_1$$. By Section 3.1, we may assume $$\mathfrak R=\mathfrak R_{ad}$$. By Corollary 4.5, it suffices to prove Theorem 2.1 (2) for $$(\tilde W_2, \mu_2, \sigma_2)$$, where $$\sigma_2$$ is a superbasic element in $$\tilde W_2$$. By Section 3.1 again, we may assume that the root datum of $$\tilde W_2$$ is adjoint. By Section 4.1, we may assume furthermore that $$\tilde W_2=\tilde W_3^m$$, where $$\tilde W_3$$ is the extended affine Weyl group of an adjoint root datum of type $$A$$ and $$\sigma_2$$ acts transitively on the set of affine simple reflections of $$\tilde W_2$$. By Lemma 5.1, it suffices to prove Theorem 2.1 (2) for $$(\tilde W_3, \mu_3, \sigma_3)$$, where $$\sigma_3=\sigma_2^m$$ is a superbasic element in $$\tilde W_3$$. This case follows from Proposition 6.1. 6.3 We recall the definition of $$\underline {\mathbf a}$$-sequence and $$\chi_{m, n}$$ in [6, Sections 3 and 5]. For $$i, j \in \mathbb Z$$, we set $$[i, j]=\{k \in \mathbb Z; i \le k \le j\}$$. Let $$r \in \mathbb N$$ and $$\chi \in \mathbb Z^r$$. For each $$j \in [1, r]$$ we define $$\underline {\mathbf a}_\chi^j: \mathbb Z_{\geq 0} \to \mathbb Z$$ by $$\underline {\mathbf a}_\chi^j(k)=\chi(j-k)$$. Here we identify $$l$$ with $$l+r$$ for $$l \in \mathbb Z$$. We say $$i \geq_\chi j$$ if $$\underline {\mathbf a}_\chi^i \geq \underline {\mathbf a}_\chi^j$$ in the sense of lexicographic order. If $$\ge_\chi$$ is a linear order, we define $$\epsilon_\chi \in \mathfrak{S}_r$$ such that $$\epsilon_\chi(i) < \epsilon_\chi(j)$$ if and only if $$i >_\chi j$$. Let $$s \leq r$$ be two nonnegative integers which are co-prime. Define $$\chi_{s, r} \in \mathbb Z^r$$ by $$\chi_{s, r}(i)=\lfloor \frac{is}{r} \rfloor-\lfloor (i-1) \frac{s}{r} \rfloor$$ for $$i \in [1, r]$$. Set $$\epsilon_{s, r}=\epsilon_{\chi_{s, r}}$$, which is well defined since $$s$$ and $$r$$ are co-prime. The following explicit description of $$\geq_{\chi_{m,n}}$$ and its application to the proof of Lemma 6.2 below are kindly suggested by one of the anonymous referees. First we identify $$[1, n]$$ with $$\mathbb Z / n \mathbb Z$$ in the natural way. Then one checks by definition that   χm,n(i)={1, if mi∈[0,m−1]⊆Z/nZ0, otherwise.  (a) Since $$m$$ is co-prime to $$n$$, $$m$$ is invertible in $$\mathbb Z / n \mathbb Z$$. We claim that   n=m−1⋅0>χm,nm−1⋅1>χm,n⋯>χm,nm−1⋅(n−1). (b) Indeed, if $$m^{-1} \cdot j_0 <_{\chi_{m,n}} m^{-1} \cdot (j_0+1)$$ for some $$j_0 \in [0, n-2]$$, there exists $$0 \leq k \leq n-1$$ such that $$\chi_{m,n}(m^{-1} \cdot j_0-i) = \chi_{m,n}(m^{-1} \cdot (j_0+1)-i)$$ for $$0 \leq i \leq k-1$$ and $$\chi_{m,n}(m^{-1} \cdot j_0-k) < \chi_{m,n}(m^{-1} \cdot (j_0+1)-k)$$. In particular, we have $$j_0+1-km=0 \in \mathbb Z / n\mathbb Z$$ by (a). Thus $$k \geq 1$$ and   χm,n(m−1⋅j0−k+1)=1>0=χm,n(m−1⋅(j0+1)−k+1), which is a contradiction. So (b) is proved. Now it is easy to see that $$\epsilon_{m,n}$$ is the permutation on $$[1, n]=\mathbb Z / n\mathbb Z$$ given by   ϵm,n(i)=mi+1 for i∈[1,n]. (c) 6.4 Let $$\mathcal S=\cup_{1 \le i \le j}\mathbb Z^{[i, j]}$$, whose elements are called segments. Let $$\eta \in \mathcal S$$ be a segment. Assume $$\eta \in \mathbb Z^{[i, j]}$$. We call $$\text{h}(\eta)=i$$ and $$\tilde \tau(\eta)=j$$ the head and the tail of $$\eta$$ , respectively. We call the positive integer $$j-i+1$$ the size of $$\eta$$. We set   |η|=∑k=h(η)t(η)η(k),av(η)=1t(η)−h(η)+1|η|. Let $$[i', j'] \subseteq [i, j]$$ be a sub-interval, we call the restriction $$\eta|_{[i',j']}$$ of $$\eta$$ to $$[i', j']$$ a subsegment of $$\eta$$ and write $$\eta|_i=\eta|_{[i,i]}$$. Let $$\theta$$ be an another segment such that $$\text{h}(\theta)=\tilde \tau(\eta)+1$$. We denote by $$\eta \vee \theta \in \mathbb Z^{[\text{h}(\eta), \tilde \tau(\theta)]}$$ the natural concatenation of $$\eta$$ and $$\theta$$. For $$k \in \mathbb Z$$, we denote by $$\eta[k]$$ the $$k$$-shift of $$\eta$$ defined by $$\eta[k](i)=\eta(i+k)$$. We say two segments are of the same type if they can be identified with each other up to some shift. For $$\eta \in \mathbb Q^{[1,n]}$$ , we denote by $$\text{Con}(\eta) \in \mathbb R^2$$ the convex hull of the points $$(0,0)$$ and $$(k, |\eta|_{[1, k]}|)$$ for $$k \in [1,n]$$. We say a subsegment $$\gamma$$ of $$\eta$$ is sharp if   av(γ)=max{av(γ′);γ′ is a subsegment of η with h(γ′)=h(γ)} and   av(γ)=min{av(γ′);γ′ is a subsegment of η with t(γ′)=t(γ)}. If $$\eta=\gamma^1 \vee \gamma^2 \vee \cdots \vee \gamma^s$$ with each $$\gamma^k$$ a sharp subsegment, then the points $$(0,0)$$ and $$(\tilde \tau(\gamma^i), |\gamma^1 \vee \cdots \vee \gamma^i|)$$ in $$\mathbb R^2$$ for $$i \in [1,s]$$ lie on the boundary of $$\text{Con}(\eta)$$ and their convex hull is just $$\text{Con}(\eta)$$. We call the dominant vector   sl(Con(η))=(av(γ1)∨⋯∨av(γs))∈Q[1,n] the slope sequence of $$\text{Con}(\eta)$$. Here for any $$\gamma \in \mathcal S$$, we define $$\text{av}^{(\gamma)} \in \mathbb Q^{[\text{h}(\gamma),\tilde \tau(\gamma)]}$$ by $$\text{av}^{(\gamma)}(i)=\text{av}(\gamma)$$ for $$i \in [\text{h}(\gamma),\tilde \tau(\gamma)]$$. Let $$\mu \in \mathbb Z^{[1,n]}$$ be a dominant cocharacter. Set $$\mu_{m,n}=\mu+\chi_{m,n}$$. Then   ⟨ωi,n,π(μm,n)⟩=⟨ωi,n,π(μ)⟩−(min−⌊min⌋)=⟨ωi,n,π(μ)+π(ϖm,n)⟩−⌈⟨ωi,n,π(ϖm,n)⟩⌉. According to the proof of Theorem 2.1 (1), the slope sequence   ν=sl(Con(π(μm,n)))=π(sl(Con(μm,n)) is the unique maximal Newton point of $$B(\tilde W_{ad}, \pi(\mu), \pi(\sigma_{m,n}))$$. 6.5 Similar to [6, Section 5], we use the Euclidean algorithm to give a recursive construction of $$\chi_{m,n}$$, which plays a crucial role in the proof of Proposition 6.1. Let $$D=\{(m,n) \in \mathbb Z_{>0}^2; m<n \text{ are co-prime}\}$$. We define $$f: D \to D \sqcup \{(1,1), (0,1)\}$$ by   f(m,n)={(m(⌊nm⌋+1)−n,m), if nm≥2;(n−(n−m)⌊nn−m⌋,n−m), otherwise. Define two types of segments $$1_{m,n}$$ and $$0_{m,n}$$ by   1m,n={(0(⌊nm⌋−1),1), if nm≥2;(0,1(⌊nn−m⌋)), otherwise,0m,n={(0(⌊nm⌋),1), if nm≥2;(0,1(⌊nn−m⌋−1)), otherwise, where the superscript $$^{(k)}$$ means to repeat the entry $$k$$ times. Here we do not fix the head or tail of these segments yet, as we are going to apply various shifts to them below. Set $$\mathcal S_1=\{\eta \in \mathcal S; \eta(i) \in \{0, 1\} \text{ for } i \in [\text{h}(\eta), \tilde \tau(\eta)]\}$$. For $$\eta \in \mathcal S_1$$ and $$k \in [\text{h}(\eta), \tilde \tau(\eta)]$$, set   η(k)m,n={1m,n, if η(k)=1;0m,n, if η(k)=0. For $$k \in [\text{h}(\eta), \tilde \tau(\eta)]$$, let $$\eta_{m,n,k}$$ be a shift of $$\eta(k)_{m,n}$$ whose head is determined recursively as follows:   h(ηm,n,k)={h(η), if k=h(η);t(ηm,n,k−1)+1, if k>h(η). Now we define $$\phi_{m,n}: \mathcal S_1 \to \mathcal S_1$$ by $$\phi_{m,n}(\eta)=\eta_{m,n,\text{h}(\eta)} \vee \cdots \vee \eta_{m,n,\tilde \tau(\eta)}$$ for $$\eta \in \mathcal S_1$$. If $$f^{h-1}(m,n) \in D$$, we set $$\phi_{m,n,h}=\phi_{m,n} \circ \cdots \circ \phi_{f^{h-1}(m,n)}$$. Using the Euclidean algorithm, one checks that   ϕm,n,h(χfh(m,n))=χm,n. We say a subsegment $$\gamma$$ of $$\chi_{m,n}$$ is of level$$h$$ if it is the image of some subsegment $$\gamma^h$$ of $$\chi_{f^h(m,n)}$$ under the map $$\phi_{m,n,h}$$. When $$h=1$$ and $$\gamma^h$$ is of size one, we say $$\gamma$$ is an elementary subsegment of $$\chi_{m,n}$$. Let $$\beta^1$$ and $$\gamma^1$$ be two segments of $$\chi^1=\chi_{f(m,n)}$$ and let $$\gamma$$ be a level one subsegment of $$\chi=\chi_{m,n}$$. Using the Euclidean algorithm, we have the following basic facts: (a) $$\text{av}(\beta^1) \geq \text{av}(\gamma^1)$$ if and only if $$\text{av}(\phi_{m,n}(\beta^1)) \geq \text{av}(\phi_{m,n}(\gamma^1))$$. (b) Each sharp subsegment of $$\gamma$$ with the same head is of level one. (c) If, moreover, $$\gamma$$ is an elementary subsegment of $$\chi$$, then $$\underline {\mathbf a}_\chi^{j} < \underline {\mathbf a}_\chi^{\text{h}(\gamma)-1}$$ and $$\underline {\mathbf a}_\chi^{j} < \underline {\mathbf a}_\chi^{\tilde \tau(\gamma)}$$ for $$j \in [\text{h}(\gamma), \tilde \tau(\gamma)-1]$$. (d) $$\underline {\mathbf a}_{\chi^1}^i < \underline {\mathbf a}_{\chi^1}^j$$ if and only if $$\underline {\mathbf a}_\chi^{\tilde \tau(\phi_{m,n}(\chi^1 |_i))} < \underline {\mathbf a}_\chi^{\tilde \tau(\phi_{m,n}(\chi^1 |_j))}$$. (e) $$\epsilon_{m,n}(n)=1$$. 6.6 Proof of Proposition 6.1 For a sequence of (distinct) elements $$i_1, i_2, \dots, i_r$$ in $$[1, n]$$, we denote by $$\text{cyc}(i_1, i_2, \dots, i_r) \in \mathfrak{S}_n$$ the cyclic permutation $$i_1 \mapsto i_2 \mapsto \cdots \mapsto i_r \mapsto i_1$$, which acts trivially on the remaining elements of $$[1,n]$$. For $$\eta \in \mathcal S$$ we set   xη=cyc(h(η),h(η)+1,…,t(η))∈S=∪i=1∞Si. Similarly, for a sequence $$\textbf{c}=(c^1, \dots, c^s)$$ of segments, we set $$x_{\textbf{c}}=x_{c^1, \dots, c^s}=x_{c^1} \cdots x_{c^s}$$. If $$\eta=c^1 \vee \cdots \vee c^s$$, we say $$\textbf{c}$$ is a decomposition of $$\eta$$. Now we are ready to prove Proposition 6.1. Write $$\chi=\chi_{m,n}$$, $$\theta=\mu_{m,n}=\mu+\chi$$ and $$\epsilon_=\epsilon_{m,n}$$. For $$h \in \mathbb Z_{>0}$$ we set $$\phi_h=\phi_{m,n,h}$$ and $$\chi^h=\chi_{f^h(m,n)}$$. By § 6.4, we have $$\nu=\pi(\text{sl}(\text{Con}(\theta)))$$. The proof will proceed as follows. First we construct a suitable sharp decomposition $$\textbf{c}$$ of $$\theta$$. One checks directly $$\pi(\nu_{w_{\textbf{c}}, \text{id}})=\pi(\text{sl}(\text{Con}(\theta)))=\nu$$, where $$w_{\textbf{c}}=t^\theta x_{\textbf{c}} \in \tilde W$$. Then we show that   ϵwcϵ−1≤ϵtθxθϵ−1=tϵ(μ)σm,n, where the last equality follows from Lemma 6.2 below. Set $$\tilde W=\epsilon w_{\textbf{c}} \epsilon^{-1} \sigma_{m, n}^{-1}$$. Then $$\tilde W \,{\le}\, t^{\epsilon(\mu)}$$ and $$\pi(\nu_{\tilde W, \sigma_{m, n}})\,{=}\,\pi(\nu_{\epsilon w_{\textbf{c}} \epsilon^{-1}, id})\,{=}\,\epsilon(\nu)$$. This completes the proof of Proposition 6.1. Lemma 6.2. We have $$\sigma_{m,n}=\epsilon t^\chi x_\chi \epsilon^{-1}$$. □ Proof We use the explicit descriptions of $$\geq_\chi$$ and $$\epsilon$$ in § 6.3 via the identification of $$[1, n]$$ with $$\mathbb Z / n\mathbb Z$$ in the natural way. Then $${u_{1,n}}=x_\chi$$ is the permutation $$i \mapsto i+1$$ on $$\mathbb Z / n\mathbb Z$$. Since $$\sigma_{m,n}=\sigma_{1,n}^m$$, we have $$u_{m,n}(i)=i+m$$ for $$i \in \mathbb Z / n\mathbb Z$$. On the other hand, we already know that $$\epsilon(i)=mi+1$$ for $$i \in \mathbb Z / n\mathbb Z$$. Therefore, $$\epsilon x_\chi \epsilon^{-1}$$ is the permutation $$mi+1 \mapsto m(i+1)+1$$ on $$\mathbb Z / n\mathbb Z$$, which equals $$u_{m,n}$$ as desired. It remains to show $$\epsilon(\chi)=\varpi_{m,n}$$. For $$1 \leq i < j \leq n$$ we have $$\epsilon(\chi)(i)=\chi(m^{-1} \cdot (i-1)) \geq \chi(m^{-1} \cdot (j-1))=\epsilon(\chi)(j)$$ since $$m^{-1} \cdot (i-1) >_{\chi} m^{-1} \cdot (j-1)$$. Thus $$\epsilon(\chi)$$ is dominant and equals $$\varpi_{m,n}$$. The proof is finished. ■ Assume $$I(\mu)=\{j \in [1,n-1]; \langle\alpha_j, \mu\rangle \neq 0\}=\{b_1, b_2,\dots, b_{r-1}\}$$ with $$b_1 < b_2 < \cdots < b_{r-1}$$. We set $$b_0=0$$ and $$b_r=n$$. Set $$\theta^i=\theta|_{[b_{i-1}+1,b_i]}$$ for $$i \in [1,r]$$. Then $$\theta=\theta^1 \vee \cdots \vee \theta^r$$. Suppose we have a sharp decomposition $$\textbf{c}_i$$ of $$\theta^i$$ for $$i \in [1, r]$$. Since $$\chi \in \{0,1\}^{[1,n]}$$ and $$\theta=\mu+\chi$$, for any subsegment $$\eta^i$$ (respectively $$\eta^j$$) of $$\theta^i$$ (respectively $$\theta^j$$) we have $$\text{av}(\eta^i) \geq \text{av}(\eta^j)$$ if $$i<j$$. Therefore the natural union $$\textbf{c}=\textbf{c}_1 \vee \cdots \vee \textbf{c}_r$$ forms a sharp decomposition of $$\theta$$. Let $$1 \leq i \leq r$$. We will construct inductively the subsegments $$\zeta_i^j, \gamma_i^j, \xi_i^j$$ for $$j \in [1, l_i]$$ (some of them might be empty) such that (a) $$\gamma^0_i=\chi |_{[b_{i-1}+1, b_j]}$$ and $$\gamma_i^{j-1}=\zeta_i^j \vee \gamma_i^j \vee \xi_i^j$$ for $$j \in [1, l_i]$$; (b) $$\zeta_i^j$$ and $$\xi_i^j$$ are sharp subsegments of $$\gamma_i^{j-1}$$; any sharp subsegment of $$\gamma_i^j$$ is also a sharp subsegment of $$\gamma_i^{j-1}$$; $$\gamma_i^{l_i}$$ is a sharp subsegment of itself (self-sharp); (c) For any$$j$$, $$\epsilon z_{i,j-1} \epsilon^{-1} \ge \epsilon z_{i,j} \epsilon^{-1}$$. Here   zi,j=tθyi−1xijvi,j;yi=xc1⋯xci;xij=xζi1,…,ζij,ξij,…,ξi1;vi,j=xγijxθi+1∨⋯∨θrcyc(t(γij),n))=cyc(h(γij),…,t(γij),bi+1,…,n). Assume we have (a), (b), and (c) for all $$i$$ and $$j$$. Set $${\zeta'}_i^j=\theta |_{[\text{h}(\zeta_i^j),\tilde \tau(\zeta_i^j)]}$$, $${\gamma'}_i^{l_i}=\theta |_{[\text{h}(\gamma_i^{l_i}),\tilde \tau(\gamma_i^{l_i})]}$$ and $${\xi'}_i^j=\theta |_{[\text{h}(\xi_i^j), \tilde \tau(\xi_i^j)]}$$. Then   ci=(ζ′i1,ζ′i2,…,ζ′ili,γ′ili,ξ′ili,…,ξ′i2,ξ′i1) forms a sharp decomposition of $$\theta^i$$, and   ϵtθxθϵ−1=ϵz1,0ϵ−1≥⋯≥ϵz1,l1+1ϵ−1=ϵz2,0ϵ−1≥⋯≥ϵzr,lr+1ϵ−1=ϵwcϵ−1 as desired. The construction is as follows. Suppose for $$1 \leq k < i$$ and $$0 \leq l \leq j$$, $$\textbf{c}_k$$, $$z_i^l$$, $$\xi_i^l$$, $$\gamma_i^l$$ are already constructed, and moreover $$\epsilon z_{i,j-1} \epsilon^{-1} \ge \epsilon z_{i,j} \epsilon^{-1}$$. We construct $$\zeta_i^{j+1}, \gamma_i^{j+1}, \xi_i^{j+1}$$ and show that $$\epsilon z_{i,j} \epsilon^{-1} \ge \epsilon z_{i, j+1} \epsilon^{-1}$$. If $$\gamma_i^j$$ is empty, there is nothing to do. Otherwise, we assume $$\gamma_i^j$$ is of level $$h$$ but not of level $$h+1$$. Then $$\gamma_i^j=\phi_h(\iota)$$ for some subsegment $$\iota$$ of $$\chi^h$$. Case (I): $$\iota$$ is not a subsegment of any elementary subsegment of $$\chi^h$$. Then there exist unique subsegments $$\zeta$$, $$\gamma$$ and $$\xi$$ of $$\chi^h$$ such that $$\gamma$$ is of level one, $$\zeta$$ (respectively $$\xi$$) is a proper subsegment of some elementary segment of $$\chi^h$$ with the same tail (respectively head), and $$\iota=\zeta \vee \gamma \vee \xi$$. Notice that at least two of $$\zeta$$, $$\gamma$$ and $$\xi$$ are nonempty. Define $$\zeta_i^{j+1}=\phi_h(\zeta)$$, $$\gamma_i^{j+1}=\phi_h(\gamma)$$ and $$\xi_i^{j+1}=\phi_h(\xi)$$. Note that $$\text{av}(\chi^h|_{[\text{h}(\zeta), \tilde \tau(\zeta)]})$$ is maximal among all subsegments of $$\chi^h$$ with the same head and $$\text{av}(\chi^h|_{[\text{h}(\xi), \tilde \tau(\xi)]})$$ is minimal among all subsegments of $$\chi^h$$ with the same tail. Therefore, (b) follows from Section 6.5 (a) and (b). To prove (c), it suffices to show that   ϵzi,j+1ϵ−1≤ϵ zi,j cyc(n,t(ζij+1)) ϵ−1; (d)  ϵ zi,j cyc(n,t(ζij+1)) ϵ−1≤ϵzi,jϵ−1. (e) Note that   ϵzi,j+1ϵ−1={ϵ zi,j cyc(n,t(ζij+1)) cyc(h(ξij+1)−1,t(ξij+1)) ϵ−1, if γ≠∅;ϵ zi,j cyc(n,t(ζij+1)) cyc(n,t(ξij+1)) ϵ−1, otherwise. Here we take $$\text{cyc}(n, \tilde \tau(\zeta_i^{j+1}))$$ (respectively $$\text{cyc}(\text{h}(\xi_i^{j+1})-1, \tilde \tau(\xi_i^{j+1}))$$ and $$\text{cyc}(n, \tilde \tau(\xi_i^{j+1}))$$) to be the identity element of $$\mathfrak{S}_n$$ if $$\zeta$$ (respectively $$\xi$$) is empty, in which case the inequality (e) (respectively (d)) becomes a priori an equality. Now we prove (d). We suppose $$\xi$$ is nonempty. Otherwise, there is nothing to prove. First we assume $$\gamma \neq \emptyset$$. By § 6.5 (c), we have $$\underline {\mathbf a}_{\chi^h}^{\text{h}(\xi)-1} > \underline {\mathbf a}_{\chi^h}^{\tilde \tau(\xi)}$$. Hence by Section 6.5 (d), $$\epsilon(\text{h}(\xi_i^{j+1})-1) < \epsilon(\tilde \tau(\xi_i^{j+1}))$$ and $$\alpha=\epsilon(e_{\text{h}(\xi_i^{j+1})-1} - e_{\tilde \tau(\xi_i^{j+1})})$$ is a positive root. Then (d) is equivalent to the following inequality   ⟨α,ϵzi,j+1−1ϵ−1(a)⟩=⟨α,ϵ(vi,jxijyi−1)−1ϵ−1(a−ϵ(θ))⟩=⟨ϵ(vi,jxijyi−1)ϵ−1(α),a−ϵ(θ)⟩=−⟨ϵ(ebi+1−eh(ξij+1)),ϵ(θ)⟩+⟨ϵ(ebi+1−eh(ξij+1)),a⟩=θ(h(ξij+1))−θ(bi+1)+⟨ϵ(ebi+1−eh(ξij+1)),a⟩>0, where $$\mathbf a$$ is the base alcove defined in Section 2.2. Note that $$1 > \langle\epsilon(e_{b_i+1}-e_{\text{h}(\xi_i^{j+1})}), \mathbf a\rangle > -1$$. Therefore, to prove (d), we have to show either $$\theta(\text{h}(\xi_i^{j+1})) > \theta(b_i+1)$$ or $$\theta(\text{h}(\xi_i^{j+1})) = \theta(b_i+1)$$ and $$\epsilon(b_i+1) <\epsilon(\text{h}(\xi_i^{j+1}))$$. Note that we always have $$\theta(\text{h}(\xi_i^{j+1}))\geq \theta(b_i+1)$$. If $$\theta(\text{h}(\xi_i^{j+1})) = \theta(b_i+1)$$, then $$\chi(b_i+1)=1>0=\chi(\text{h}(\xi_i^{j+1}))$$. Hence $$\epsilon(b_i+1) <\epsilon(\text{h}(\xi_i^{j+1}))$$ as desired. Now we assume $$\gamma= \emptyset$$. Note that $$\epsilon(n) < \epsilon(\tilde \tau(\xi_i^{j+1}))$$. Then (d) follows by a similar argument as in the case of $$\gamma \neq \emptyset$$ with $$\text{h}(\xi_i^{j+1})-1$$ replaced by $$n$$. To prove (e), again we suppose that $$\zeta$$ is nonempty. Since one of $$\gamma$$ and $$\xi$$ is nonempty, $$\tilde \tau(\zeta_i^{j+1}) \neq n$$. By Section 6.5 (e), $$1=\epsilon(n) < \epsilon(\tilde \tau(\zeta_i^{j+1}))$$. As in the proof of (d), we see that (e) holds if $$\theta(\tilde \tau(\zeta_i^{j+1})+1) < \theta(\text{h}(\zeta_i^{j+1}))$$. Otherwise, we have $$\theta(\tilde \tau(\zeta_i^{j+1})+1) = \theta(\text{h}(\zeta_i^{j+1}))$$, which implies that $$\chi(\text{h}(\zeta_i^{j+1}))=\chi(\tilde \tau(\zeta_i^{j+1})+1)=0$$. Since $$\zeta$$ is a proper subsegment of some elementary segment of $$\chi^h$$ and shares the same tail with it, by Section 6.5 (c) we have that $$\underline {\mathbf a}_{\chi^h}^{\tilde \tau(\zeta)} > \underline {\mathbf a}_{\chi^h}^{\text{h}(\zeta)-1}$$. Hence by § 6.5 (d) and that $$\chi(\tilde \tau(\zeta_i^{j+1})+1)=\chi(\text{h}(\zeta_i^{j+1}))=0$$, we have $$\underline {\mathbf a}_\chi^{\tilde \tau(\zeta_i^{j+1})+1} > \underline {\mathbf a}_\chi^{\text{h}(\zeta_i^{j+1})}$$. So $$\epsilon(\text{h}(\zeta_i^{j+1})) > \epsilon(\tilde \tau(\zeta_i^{j+1})+1)$$ and (e) holds. Case (II): $$\iota$$ is a subsegment of some elementary subsegment of $$\chi^h$$. We define $$l_i=j$$ and the construction of $$\textbf{c}_i$$ is finished. One checks directly that $$\iota$$ is self-sharp, hence so is $$\gamma_i^{l_i}=\phi_h(\iota)$$ by Section 6.5 (a) and (b). If $$\tilde \tau(\gamma_i^{l_i})=n$$, the induction step is finished. Otherwise, it remains to show   ϵzi,liϵ−1≥ϵ zi,li cyc(t(γili),n) ϵ−1=ϵzi,li+1ϵ−1. (f) Note that $$1=\epsilon(n) < \epsilon(\tilde \tau(\gamma_i^{l_i}))$$. Again, we see that (f) holds if $$\theta(\text{h}(\gamma_i^{l_i})) > \theta(b_i+1)$$. Otherwise, we have $$\theta(\text{h}(\gamma_i^{l_i})) = \theta(b_i+1)$$, $$\chi(\text{h}(\gamma_i^{l_i}))=0$$ and $$\chi(b_i+1)=1$$ since $$b_i \in I(\mu)$$. Hence $$\epsilon(\text{h}(\gamma_i^{l_i})) > \epsilon(b_i+1)$$ and (f) still holds. Example 6.3. Finally we provide an example. Let $$n=8$$, $$m=5$$ and $$\mu=(1, 1, 1, 0, 0, 0, 0, 0)$$. Then $$\chi_{m,n}=(0, 1, 0, 1, 1, 0, 1, 1) \in \mathbb Z^8$$, $$\epsilon_{m,n}=\text{cyc}(1, 6, 7, 4, 5, 2, 3, 8)$$ and $$u_{m, n}=\text{cyc}(6, 3, 8, 5, 2, 7, 4, 1)$$. Note that $$I(\mu)=3$$. Applying the algorithm in the proof of Proposition 6.1, we obtain the following sharp decomposition:   μm,n=(1,2,1,1,1,0,1,1)=(1,2)∨(1)∨(1,1)∨(0,1,1). Hence $$\nu=(\frac{1}{2}, \frac{1}{2}, 0, 0, 0, -\frac{1}{3}, -\frac{1}{3}, -\frac{1}{3})$$. Moreover, one checks that   tϵm,n(μ)σm,n≥tϵm,n(μ)σm,ncyc(8,3)≥tϵm,n(μ)σm,ncyc(8,3)cyc(1,3)≥tϵm,n(μ)σm,ncyc(8,3)cyc(1,3)cyc(1,2). Set $$\tilde W=t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3) \text{cyc}(1, 3) \text{cyc}(1, 2) \sigma_{m, n}^{-1}$$. Then $$\tilde W \le t^{\epsilon_{m, n}(\mu)}$$ and $$\pi(\bar \nu_{\tilde W, \sigma_{m,n}})=\nu$$. This verifies Proposition 6.1 in this case. □ Acknowledgement We thank the referees for their careful reading of this paper and many useful comments. Appendix: The Set $$B(G, \{\mu\})$$ In the appendix, we discuss the relation between the set $$B(\tilde W, \mu, \sigma)$$ defined in Section 2.4 and the set $$B(G, \mu)$$ for $$p$$-adic groups. A.1 Recall that $$F$$ is a finite field extension of $$\mathbb Q_p$$, $$L$$ is the completion of the maximal unramified extension of $$F$$ and $$G$$ is a connected reductive algebraic group over $$F$$. We first discuss the Iwahori-Weyl group of $$G$$ over $$L$$. We follow [5]. Let $$S$$ be a maximal $$L$$-split torus that is defined over $$F$$ and let $$T$$ be its centralizer. Since $$G$$ is quasi-split over $$L$$, $$T$$ is a maximal torus. Let $$N$$ be the normalizer of $$T$$. The finite Weyl group associated to $$S$$ is $$W_0=N(L)/T(L).$$ The Iwahori–Weyl group associated to $$S$$ is $$\tilde W=N(L)/T(L)_1$$, where $$T(L)_1$$ denotes the unique Iwahori subgroup of $$T(L)$$. Let $$\Gamma=\text{Gal}(\bar L/L)$$. As in [5, p. 195] and [15, Section 4.2], one associates a reduced root system $$R$$ to $$(G, T)$$. Let $$V=X_*(T)_\Gamma \otimes \mathbb R=X_*(S) \otimes \mathbb R$$. We fix a $$\sigma$$-invariant alcove $$\mathbf a_G$$ in the apartment of $$S$$ along with a special vertex of $$\mathbf a_G$$. The special vertex allows us to identify $$V$$ with the apartment of $$S$$. The alcove $$\mathbf a_G$$ is contained in a unique (relative) Weyl chamber of $$V$$, which we call the dominant chamber. The hyperplanes in $$V$$ then give a reduced root system $$R$$. We have the semi-direct product   W~=X∗(T)Γ⋊W0. The group $$X_*(T)_\Gamma$$ is not torsion-free in general. However, by [4, §8.1], the torsion part $$X_*(T)_{\Gamma, tor}$$ lies in the center of $$\tilde W$$ and one may identify the extended affine Weyl group of $$\mathfrak R$$ with $$\tilde W/X_*(T)_{\Gamma, tor}$$. For our purpose, it suffices to consider the case where $$X_*(T)_\Gamma$$ is torsion-free. In this case, the root system $$R$$, together with the cocharacter group $$X_*(T)_\Gamma$$, defines a reduced datum $$\mathfrak R$$, and $$\tilde W$$ is the extended affine Weyl group of $$\mathfrak R$$ introduced in Section 2. The Iwahori-Weyl group $$\tilde W$$ contains the affine Weyl group $$W_a$$ as a normal subgroup and   W~=Wa⋊Ω, where $$\Omega \cong \pi_1(G)_{\Gamma}$$ is the normalizer of the alcove $$\mathbf a_G$$. The Bruhat order on $$W_a$$ extends in a natural way to $$\tilde W$$. The Frobenius morphism $$\sigma$$ induces an action on $$\tilde W$$, which we denote by $$\text{Ad}(\sigma)$$. A.2 Recall that $$\{\mu\}$$ is a geometric conjugacy class of cocharacters of $$G$$. We may regard $$\{\mu\}$$ as a conjugacy class in $$X_*(T)$$ under the absolute Weyl group. Following [15, Section 4.3], let $$\tilde \Lambda_{\{\mu\}} \subseteq \{\mu\}$$ be the subset of cocharacters which are $$B$$-dominant for some Borel subgroup $$B$$ defined over $$L$$ with $$B \supseteq T$$. Then $$\tilde \Lambda_{\{\mu\}}$$ is a single $$W_0$$-orbit. Let $$\Lambda_{\{\mu\}}$$ be the image of $$\tilde \Lambda_{\{\mu\}}$$ in $$X_*(T)_\Gamma$$. The $$\{\mu\}$$-admissible set is defined by   Adm({μ})={w∈W~;w≤tξ for some ξ∈Λ{μ}}. Let $$\tilde \mu$$ be the unique dominant cocharacter in $$\tilde\Lambda_{\{\mu\}}$$ and $$\mu$$ be its image in $$\Lambda_{\{\mu\}}$$. Then $$\Lambda_{\{\mu\}}=W_0 \cdot \mu$$ and $$\text{Adm}(\{\mu\})$$ equals $$\text{Adm}(\mu)$$ defined in § 2.4. (In the function field case, it is shown in [17, Remark 2.11] that $$\text{Adm}(\{\mu\})=\{w \in \tilde W; w \le t^\xi \text{ for some } \xi \in im\{\mu\} \subseteq X_*(T)_\Gamma\}$$. We do not need this result here. A.3 The set $$B(G)$$ of $$\sigma$$-conjugacy classes of $$G(L)$$ is classified by Kottwitz in [11] and [12]. For any $$b \in G(L)$$, we denote by $$[b]$$ the $$\sigma$$-conjugacy class of $$G(L)$$ that contains $$b$$. Let $$\Gamma_F=\text{Gal}(\bar L/F)$$ be the absolute Galois group of $$F$$. Let $$\kappa_G: B(G) \to \pi_1(G)_{\Gamma_F}$$ be the Kottwitz map [12, § 7]. This gives one invariant. Another invariant is given by the Newton map. To an element $$b \in G(L)$$, we associate its Newton point $$\bar \nu_b \in X_*(T)_{\Gamma} \otimes \mathbb R$$. By [12, Section 4.13], the map   B(G)→π1(G)ΓF×(X∗(T)Γ⊗R),b↦(κG(b),ν¯b) is injective. For any $$w \in \tilde W$$, we choose a representative in $$N(L)$$ and also write it as $$w$$. The map $$N(L) \to G(L)$$ induces a map $$\tilde W \to B(G)$$. By [7, Section 3] and [3, 2.4], this map is surjective. ([3, 2.4] is stated for adjoint groups, but the result for arbitrary groups holds by combining with the reduction argument in [3, 2.3].) The restrictions of the Kottwitz map and the Newton map on $$\tilde W \subseteq G(L)$$ are the maps $$\kappa_{\tilde W, \sigma}$$ and $$w \mapsto \bar \nu_{w, \sigma}$$ defined in § 2.3. A.4 In this section, we assume furthermore that $$G$$ is a quasi-split connected reductive group over $$F$$ and that $$H$$ is an inner form of $$G$$. We denote by $$\sigma_G$$ and $$\sigma_H$$ the Frobenius morphisms of $$G$$ and $$H$$ , respectively. Via the canonical isomorphism $$X_*(T)_{\Gamma} \otimes \mathbb R \cong (X_*(T) \otimes \mathbb R)^\Gamma$$, we may identify $$\mu^\diamondsuit_{\sigma_G}$$ in § 2.4 with $$[\Gamma_F: \Gamma_{F, \tilde \mu}]^{-1} \sum_{\tau \in \Gamma_F/\Gamma_{F, \tilde \mu}} \tau(\tilde \mu)$$ in [12, (6.1.1)], where $$\Gamma_{F, \tilde \mu}$$ is the isotropy group of $$\mu$$ in $$\Gamma_F$$. By Lemma 3.4, $$\mu^\diamondsuit_{\sigma_G}=\mu^{\diamondsuit}_{\sigma_H}$$. Let $$\mu^\sharp$$ be the image of $$\tilde \mu$$ under the natural map $$X_*(T) \to \pi_1(H)_{\Gamma_F}$$. Set   B(H,{μ})={[b]∈B(H);κH(b)=μ♯,ν¯b≤μσH♢}. Then we may identify $$B(H, \{\mu\})$$ with $$B(\tilde W, \mu, \sigma_H)$$. Theorem 2.1 may be reformulated as follows: The set $$B(H, \{\mu\})$$ contains a unique maximal element and this element is represented by an element in $$\text{Adm}(\{\mu\})$$. References [1] Bourbaki N. Groupes et algèbres de Lie, Ch. 4,5,6 . Paris: Hermann, 1968. [2] Chai C. “Newton polygon as lattice points.” American Journal of Mathematics  122, no. 5 ( 2000): 967– 90. Google Scholar CrossRef Search ADS   [3] Görtz U. He X. and Nie. S. “$$P$$-alcoves and nonemptiness of affine Deligne-Lusztig varieties.” Annales Scientifiques de l’École Normale SupÉrieure  48 ( 2015): 647– 65. [4] Haines T. and He. X. “Vertexwise criteria for admissibility of alcoves.” arXiv:1411.5450, to appear in The American Journal of Mathematics. [5] Haines T. and Rapoport M. “On parahoric subgroups.” Advances in Mathematics 219 ( 2008): 188– 98. [6] He. X. “Minimal length elements in conjugacy classes of extended affine Weyl groups.” arXiv: 1004.4040. [7] He X. “Geometric and homological properties of affine Deligne-Lusztig varieties.” Annals of Mathematics  179 ( 2014): 367– 404. Google Scholar CrossRef Search ADS   [8] He. X. “Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties.” arXiv:1408.5838, to appear in Annales Scientifiques de l’Ècole Normale Supérieure. [9] He X. and S. Nie. “Minimal length elements of extended affine Weyl group.” Compositio Mathematica  150 ( 2014): 1903– 27. Google Scholar CrossRef Search ADS   [10] He X. and S. Nie. “$$P$$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras.” Selecta Mathematica  21 ( 2015): 995– 1019. Google Scholar CrossRef Search ADS   [11] Kottwitz R. “Isocrystals with additional structure.” Compositio Mathematica  56 ( 1985): 201– 20. [12] Kottwitz R. “Isocrystals with additional structure. II.” Compositio Mathematica  109 ( 1997): 255– 339. Google Scholar CrossRef Search ADS   [13] Kottwitz R. and Rapoport. M. “Minuscule alcoves for $$GL_n$$ and $$GSp_{2n$$.” Selecta Mathematica  102 ( 2000): 403– 28. [14] Kottwitz R. and Rapoport M. “On the existence of F -crystals.” Commentarii Mathematici Helvetici  78 ( 2003): 153– 84. Google Scholar CrossRef Search ADS   [15] Pappas G. Rapoport M. and B. Smithling. Local Models of Shimura Varieties, I. Geometry and Combinatorics,  pp. 135– 217. Handbook of moduli vol. III, Advanced Lectures in Mathematics (ALM) vol. 26. Somerville, MA: Int. Press, 2013. [16] Rapoport M. “A guide to the reduction modulo $$p$$ of Shimura varieties.” Astérisque  298 ( 2005): 271– 318. [17] Richarz T. “Affine Grassmannians and geometric satake equivalences.” arXiv:1311.1008. [18] Springer T. A. “Regular elements of finite reflection groups.” Inventiones Mathematicae  25 ( 1974): 159– 98. Google Scholar CrossRef Search ADS   © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

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International Mathematics Research NoticesOxford University Press

Published: Feb 1, 2018

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