Characters of Integrable Highest Weight Modules over a Quantum Group

Characters of Integrable Highest Weight Modules over a Quantum Group Abstract We show that the Weyl–Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac–Moody Lie algebra, when the parameter $$q$$ is not a root of unity. 1 Introduction It is well-known that the character of an integrable highest weight module over a symmetrizable Kac–Moody algebra $${\mathfrak{g}}$$ is given by the Weyl–Kac character formula [6]. In this paper we consider the corresponding problem for a quantized enveloping algebra [7]. For a field $$K$$ and $$z\in K^\times$$ which is not a root of 1, we denote by $$U_{K,z}({\mathfrak{g}})$$ the quantized enveloping algebra of $${\mathfrak{g}}$$ over $$K$$ at $$q=z$$, namely the specialization of Lusztig’s $${\mathbb{Z}}[q,q^{-1}]$$-form via $$q\mapsto z$$. It is already known that the Weyl–Kac type character formula holds for $$U_{K,z}({\mathfrak{g}})$$ in some cases. When $$K$$ is of characteristic $$0$$ and $$z$$ is transcendental, this is due to Lusztig [10]. When $${\mathfrak{g}}$$ is finite-dimensional, this is shown in [1]. When $${\mathfrak{g}}$$ is affine, this is known in certain specific cases [2, 15]. We first point out that the problem is closely related to the non-degeneracy of the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$. In fact, assume we could show that the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ is non-degenerate. Then we can define the quantum Casimir operator. It allows us to apply Kac’s argument for Lie algebras in [6] to $$U_{K,z}({\mathfrak{g}})$$, and we obtain the Weyl–Kac type character formula for integrable highest weight modules over $$U_{K,z}({\mathfrak{g}})$$. In particular, we can deduce the Weyl–Kac type character formula in the affine case from the case-by-case calculation of the Drinfeld pairing due to Damiani [3, 4]. The aim of this paper is to give a simple unified proof of the non-degeneracy of the Drinfeld pairing and the Weyl–Kac type character formula for $$U_{K,z}({\mathfrak{g}})$$, where $${\mathfrak{g}}$$ is a symmetrizable Kac–Moody algebra, $$K$$ is a field not necessarily of characteristic zero, and $$z\in K^\times$$ is not a root of 1. Our argument is as follows. We consider the (possibly) modified algebra $${\overline{U}}_{K,z}({\mathfrak{g}})$$, which is the quotient of $$U_{K,z}({\mathfrak{g}})$$ by the ideal generated by the radical of the Drinfeld pairing. Then the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ induces a non-degenerate pairing for $${\overline{U}}_{K,z}({\mathfrak{g}})$$, by which we can define the quantum Casimir operator for $${\overline{U}}_{K,z}({\mathfrak{g}})$$. It allows us to apply Kac’s argument for Lie algebras to $${\overline{U}}_{K,z}({\mathfrak{g}})$$, and we obtain the Weyl–Kac type character formula for $${\overline{U}}_{K,z}({\mathfrak{g}})$$ with modified denominator. In the special case where the highest weight is zero, this gives a formula for the modified denominator. Comparing this with the ordinary denominator formula for Lie algebras, we conclude that the modified denominator coincides with the original denominator for the Lie algebra $${\mathfrak{g}}$$. It implies that the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ was already non-degenerate. This is the outline of our argument. In applying Kac’s argument to the modified algebra, we need to show that the modified denominator is skew invariant with respect to a twisted action of the Weyl group. This is accomplished using certain standard properties of the Drinfeld pairing. 2 Quantized Enveloping Algebras Let $${\mathfrak{h}}$$ be a finite-dimensional vector space over $${\mathbb{Q}}$$, and let $$\{h_i\}_{i\in I}$$ and $$\{\alpha_i\}_{i\in I}$$ be linearly independent subsets of $${\mathfrak{h}}$$ and $${\mathfrak{h}}^*$$, respectively, such that $$(\langle \alpha_j,h_i\rangle)_{i, j\in I}$$ is a symmetrizable generalized Cartan matrix. We denote by $$W$$ the associated Weyl group. It is a subgroup of $$GL({\mathfrak{h}})$$ generated by the involutions $$s_i$$ ($$i\in I$$) defined by $$s_i(h)=h-\langle\alpha_i,h\rangle h_i$$ for $$h\in {\mathfrak{h}}$$. The contragredient action of $$W$$ on $${\mathfrak{h}}^*$$ is given by $$s_i(\lambda)=\lambda-\langle\lambda,h_i\rangle \alpha_i$$ for $$i\in I$$, $$\lambda\in {\mathfrak{h}}^*$$. Set   E=∑i∈IQαi,Q=∑i∈IZαi,Q+=∑i∈IZ≧0αi. We can take a symmetric $$W$$-invariant bilinear form $$ (\;,\;): E\times E\to{\mathbb{Q}} $$ such that   (αi,αi)2∈Z>0(i∈I). (2.1) For $$\lambda\in E$$ and $$i\in I$$ we obtain from $$(\lambda,\alpha_i)=(s_i\lambda,s_i\alpha_i)$$ that   ⟨λ,hi⟩=2(λ,αi)(αi,αi). (2.2) In particular we have   (αi,αj)=⟨αj,hi⟩(αi,αi)2∈Z, and hence $$(Q,Q)\subset{\mathbb{Z}}$$. For $$i\in I$$ set $$t_i=\frac{(\alpha_i,\alpha_i)}2h_i$$, and for $$\gamma=\sum_in_i\alpha_i\in Q$$ set $$t_\gamma=\sum_in_it_i$$. By (2.2) we have $$(\lambda,\gamma)=\langle\lambda,t_\gamma\rangle$$ for $$\lambda\in E$$, $$\gamma\in Q$$. We fix a $${\mathbb{Z}}$$-form $${\mathfrak{h}}_{\mathbb{Z}}$$ of $${\mathfrak{h}}$$ such that   ⟨αi,hZ⟩⊂Z,ti∈hZ(i∈I). (2.3) We set   P={λ∈h∗∣⟨λ,hZ⟩⊂Z},P+={λ∈P∣⟨λ,hi⟩∈Z≧0}. We fix $$\rho\in{\mathfrak{h}}^*$$ such that $$\langle\rho, h_i\rangle=1$$ for any $$i\in I$$, and define a twisted action of $$W$$ on $${\mathfrak{h}}^*$$ by   w∘λ=w(λ+ρ)−ρ(w∈W,λ∈h∗). This action does not depend on the choice of $$\rho$$, and we have $$w\circ P=P$$ for any $$w\in W$$. Denote by $${\mathcal E}$$ the set of formal sums $$\sum_{\lambda\in P}c_\lambda e(\lambda)$$; ($$c_\lambda\in {\mathbb{Z}}$$) such that there exist finitely many $$\lambda_1,\dots, \lambda_r\in P$$ such that   {λ∈P∣cλ≠0}⊂⋃k=1r(λk−Q+). Note that $${\mathcal E}$$ is naturally a commutative ring by the multiplication $$e(\lambda)e(\mu)=e(\lambda+\mu)$$. Denote by $$\Delta^+$$ the set of positive roots for the Kac–Moody Lie algebra $${\mathfrak{g}}$$ associated with the generalized Cartan matrix $$(\langle\alpha_j, h_i\rangle)_{i,j\in I}$$. For $$\alpha\in\Delta^+$$ let $$m_\alpha$$ be the dimension of the root space of $${\mathfrak{g}}$$ with weight $$\alpha$$. We define an invertible element $$D$$ of $${\mathcal E}$$ by   D=∏α∈Δ+(1−e(−α))mα. For $$n\in{\mathbb{Z}}_{\geqq0}$$ set   [n]x=xn−x−nx−x−1∈Z[x,x−1],[n]!x=[n]x[n−1]x⋯[1]x∈Z[x,x−1]. We denote by $${\mathbb{F}}={\mathbb{Q}}(q)$$ the field of rational functions in the variable $$q$$ with coefficients in $${\mathbb{Q}}$$. The quantized enveloping algebra $$U$$ associated with $${\mathfrak{h}}$$, $$\{h_i\}_{i\in I}$$, $$\{\alpha_i\}_{i\in I}$$, $${\mathfrak{h}}_{\mathbb{Z}}$$, $$(\;,\;)$$ is the associative algebra over $${\mathbb{F}}$$ generated by the elements $$k_h$$, $$e_i$$, $$f_i$$ ($$h\in{\mathfrak{h}}_{\mathbb{Z}}$$, $$i\in I$$) satisfying the relations   k0=1,khkh′=kh+h′ (h,h′∈hZ), (2.4)  kheik−h=qi⟨αi,h⟩ei (h∈hZ,i∈I), (2.5)  khfik−h=qi−⟨αi,h⟩fi (h∈hZ,i∈I), (2.6)  eifj−fjei=δijki−ki−1qi−qi−1 (i,j∈I), (2.7)   ∑r+s=1−⟨αj,hi⟩(−1)rei(r)ejei(s)=0 (i,j∈I,i≠j), (2.8)   ∑r+s=1−⟨αj,hi⟩(−1)rfi(r)fjfi(s)=0 (i,j∈I,i≠j), (2.9) where $$k_i=k_{t_i}$$, $$q_i=q^{(\alpha_i,\alpha_i)/2}$$ for $$i\in I$$, and $$e_i^{(r)}=\frac1{[r]!_{q_i}}e_i^r$$, $$f_i^{(r)}=\frac1{[r]!_{q_i}}f_i^r$$ for $$i\in I$$, $$r\in{\mathbb{Z}}_{\geqq0}$$. For $$\gamma\in Q$$ we set $$k_\gamma=k_{t_\gamma}$$. We have a Hopf algebra structure of $$U$$ given by   Δ(kh)=kh⊗kh,Δ(ei)=ei⊗1+ki⊗ei,Δ(fi)=fi⊗ki−1+1⊗fi, (2.10)  ε(kh)=1,ε(ei)=ε(fi)=0, (2.11)  S(kh)=kh−1,S(ei)=−ki−1ei,S(fi)=−fiki (2.12) for $$h\in{\mathfrak{h}}_{\mathbb{Z}}, i\in I$$. We will sometimes use Sweedler’s notation for the coproduct   Δ(u)=∑(u)u(0)⊗u(1)(u∈U), and the iterated coproduct   Δm(u)=∑(u)mu(0)⊗⋯⊗u(m)(u∈U). We define $${\mathbb{F}}$$-subalgebras $$U^0$$, $$U^+$$, $$U^-$$, $$U^{\geqq0}$$, $$U^{\leqq0}$$ of $$U$$ by   U0=⟨kh∣h∈hZ⟩,U+=⟨ei∣i∈I⟩,U−=⟨fi∣i∈I⟩,U≧0=⟨kh,ei∣h∈hZ,i∈I⟩,U≦0=⟨kh,fi∣h∈hZ,i∈I⟩. For $$\gamma\in Q$$ set   Uγ={u∈U∣khukh−1=q⟨γ,h⟩u(h∈hZ)},Uγ±=Uγ∩U±. Then we have   U0=⨁h∈hZFkh,U±=⨁γ∈Q+U±γ±. It is known that the multiplication of $$U$$ induces isomorphisms   U≅U+⊗U0⊗U−≅U−⊗U0⊗U+,U≧0≅U+⊗U0≅U0⊗U+,U≦0≅U−⊗U0≅U0⊗U− of vector spaces. It is also known that   ∑γ∈Q+dim⁡U−γ−e(−γ)=D−1. (2.13) For a $$U$$-module $$V$$ and $$\lambda\in P$$ we set   Vλ={v∈V∣khv=q⟨λ,h⟩v(h∈hZ)}. We say that a $$U$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ and $$i\in I$$ there exists some $$N>0$$ such that $$e_i^{(n)}v=f_i^{(n)}v=0$$ for $$n\geqq N$$. For $$i\in I$$ and an integrable $$U$$-module $$V$$ define an operator $$T_i:V\to V$$ by   Tiv=∑−a+b−c=⟨λ,hi⟩(−1)bqi−ac+bei(a)fi(b)ei(c)v(v∈Vλ). It is invertible and satisfies $$T_iV_\lambda= V_{s_i\lambda}$$ for $$\lambda\in P$$. There exists a unique algebra automorphism $$T_i:U\to U$$ such that for any integrable $$U$$-module $$V$$ we have $$ T_iuv=T_i(u)T_iv$$;($$u\in U, v\in V$$). Then we have $$T_i(U_\gamma)=U_{s_i\gamma}$$ for $$\gamma\in Q$$. The action of $$T_i$$ on $$U$$ is given by   Ti(kh) =ksih,Ti(ei)=−fiki,Ti(fi)=−ki−1ei(h∈hZ),Ti(ej) =∑r+s=−⟨αj,hi⟩(−1)rqi−rei(s)ejei(r)(j∈I,i≠j),Ti(fj) =∑r+s=−⟨αj,hi⟩(−1)rqirfi(r)fjfi(s)(j∈I,i≠j) [11, Section 37.1]. The multiplication of $$U$$ induces   U+≅(U+∩Ti(U+))⊗F[ei]≅F[ei]⊗(U+∩Ti−1(U+)), (2.14)  U−≅(U−∩Ti(U−))⊗F[fi]≅F[fi]⊗(U−∩Ti−1(U−)) (2.15) [11, Lemma 38.1.2]. Moreover,   Δ(U+∩Ti(U+))⊂U≧0⊗(U+∩Ti(U+)), (2.16)  Δ(U+∩Ti−1(U+))⊂U0(U+∩Ti−1(U+))⊗U+, (2.17)  Δ(U−∩Ti(U−))⊂(U−∩Ti(U−))⊗U≦0, (2.18)  Δ(U−∩Ti−1(U−))⊂U−⊗U0(U−∩Ti−1(U−)) (2.19) [13, Lemma 2.8]. Set   ♯U0=⨁γ∈QFkγ⊂U0,♯U≧0=♯U0U+,♯U≦0=♯U0U−. They are Hopf subalgebras of $$U$$. The Drinfeld pairing is the bilinear form   τ:♯U≧0×♯U≦0→F characterized by the following properties:   τ(x,y1y2)=(τ⊗τ)(Δ(x),y1⊗y2) (x∈♯U≧0,y1,y2∈♯U≦0), (2.20)  τ(x1x2,y)=(τ⊗τ)(x2⊗x1,Δ(y)) (x1,x2∈♯U≧0,y∈♯U≦0), (2.21)  τ(kγ,kδ)=q−(γ,δ) (γ,δ∈Q), (2.22)  τ(ei,fj)=−δij(qi−qi−1)−1 (i,j∈I), (2.23)  τ(ei,kγ)=τ(kγ,fi)=0 (i∈I,γ∈Q). (2.24) It satisfies the following properties:   τ(xkγ,ykδ)=τ(x,y)q−(γ,δ) (x∈U+,y∈U−,γ,δ∈Q), (2.25)  τ(Uγ+,U−δ−)={0} (γ,δ∈Q+,γ≠δ), (2.26)  τ|Uγ+×U−γ−is non-degenerate (γ∈Q+), (2.27)  τ(Sx,Sy)=τ(x,y) (x∈♯U≧0,y∈♯U≦0). (2.28) Moreover, for $$x\in{}^\sharp{U}^{\geqq0}$$, $$y\in {}^\sharp{U}^{\leqq0}$$ we have   xy=∑(x)2,(y)2τ(x(0),y(0))τ(x(2),Sy(2))y(1)x(1), (2.29)  yx=∑(x)2,(y)2τ(Sx(0),y(0))τ(x(2),y(2))x(1)y(1) (2.30) [12, Lemma 2.1.2]. For $$i\in I$$ we define linear maps   ri,±:U±→U±,ri,±′:U±→U± by   Δ(x)∈ri,+(x)ki⊗ei+∑δ∈Q+∖{αi}U≧0⊗Uδ+ (x∈U+),Δ(x)∈eikγ−αi⊗ri,+′(x)+∑δ∈Q+∖{αi}Uδ+U0⊗U+ (x∈Uγ+),Δ(y)∈ri,−(y)⊗fik−γ+αi+∑δ∈Q+∖{αi}U−⊗U−δ−U0 (y∈U−γ−),Δ(y)∈fi⊗ri,−′(y)ki−1+∑δ∈Q+∖{αi}U−δ−⊗U≦0 (y∈U−). We have   U+∩Ti(U+)= {u∈U+∣τ(u,U−fi)={0}}={u∈U+∣ri,+(u)=0}, (2.31)  U+∩Ti−1(U+)= {u∈U+∣τ(u,fiU−)={0}}={u∈U+∣ri,+′(u)=0}, (2.32)  U−∩Ti(U−)= {u∈U−∣τ(U+ei,u)={0}}={u∈U−∣ri,−′(u)=0}, (2.33)  U−∩Ti−1(U−)= {u∈U−∣τ(eiU+,u)={0}}={u∈U−∣ri,−(u)=0} (2.34) [11, Proposition 38.1.6]. By (2.16–2.19) and (2.31–2.34) we easily obtain   τ(xeim,yfin)=δmnτ(x,y)qin(n−1)/2(qi−1−qi)n[n]!qi (x∈U+∩Ti(U+),y∈U−∩Ti(U−)), (2.35)  τ(eimx′,finy′)=δmnτ(x′,y′)qin(n−1)/2(qi−1−qi)n[n]!qi (x′∈U+∩Ti−1(U+),y′∈U−∩Ti−1(U−)). (2.36) We have also   τ(x,y)=τ(Ti−1(x),Ti−1(y))(x∈U+∩Ti(U+),y∈U−∩Ti(U−)) (2.37) [11, Proposition 38.2.1], [14, Theorem 5.1]. 3 Specialization Let $$R$$ be a subring of $${\mathbb{F}}={\mathbb{Q}}(q)$$ containing $${\mathbb{A}}={\mathbb{Z}}[q,q^{-1}]$$. We denote by $$U_R$$ the $$R$$-subalgebra of $$U$$ generated by $$k_h$$, $$e_i^{(n)}$$, $$f_i^{(n)}$$ ($$h\in{\mathfrak{h}}_{\mathbb{Z}}, i\in I, n\geqq0$$). It is a Hopf algebra over $$R$$. We define subalgebras $$U_R^0$$, $$U_R^+$$, $$U_R^-$$, $$U_R^{\geqq0}$$, $$U_R^{\leqq0}$$ of $$U_R$$ by   UR0=U0∩UR,UR±=U±∩UR,UR≧0=U≧0∩UR,UR≦0=U≦0∩UR. Setting $$ U_{R,\pm\gamma}^\pm=U_{\pm\gamma}^\pm\cap U_R$$ for $$\gamma\in Q^+ $$ we have   UR±=⨁γ∈Q+UR,±γ±. It is known that $$U^\pm_{R,\pm\gamma}$$ is a free $$R$$-module of rank $$\dim U^\pm_{\pm\gamma}$$ [11, Section 14.2]. Hence we have   ∑γ∈Q+rankR(UR,−γ−)e(−γ)=D−1 (3.1) by (2.13). The multiplication of $$U_R$$ induces isomorphisms   UR≅UR+⊗UR0⊗UR−≅UR−⊗UR0⊗UR+,UR≧0≅UR+⊗UR0≅UR0⊗UR+,UR≦0≅UR−⊗UR0≅UR0⊗UR− of $$R$$-modules. For $$i\in I$$ the algebra automorphisms $$T_i^{\pm1}:U\to U$$ preserve $$U_R$$. Lemma 3.1. The multiplication of $$U_R$$ induces isomorphisms   UR+≅(UR+∩Ti(UR+))⊗R(⨁n=0∞Rei(n)), (3.2)  UR+≅(⨁n=0∞Rei(n))⊗R(UR+∩Ti−1(UR+)), (3.3)  UR−≅(UR−∩Ti(UR−))⊗R(⨁n=0∞Rfi(n)), (3.4)  UR−≅(⨁n=0∞Rfi(n))⊗R(UR−∩Ti−1(UR−)). (3.5) □ Proof. We only show (3.2). The injectivity of the canonical homomorphism   (UR+∩Ti(UR+))⊗R(⨁n=0∞Rei(n))→UR+ is clear. To show the surjectivity it is sufficient to verify that its image is stable under the left multiplication by $$e_j^{(n)}$$ for any $$j\in I$$ and $$n\geqq0$$. If $$j\ne i$$, this is clear since $$e_j^{(n)}\in U_R^+\cap T_i(U_R^+)$$. Consider the case $$j=i$$. By (2.31) and the general formula   ri,+(xx′)=qi⟨γ′,αi∨⟩ri,+(x)x′+xri,+(x′)(x∈U+,x′∈Uγ′+) we easily obtain   x∈Uγ+∩Ti(U+)⟹eix−qi⟨γ,αi∨⟩xei∈Uγ+αi+∩Ti(U+). Now let $$x\in U^+_{R,\gamma}\cap T_i(U^+_{R})$$. Define $$x_k\in U^+_{\gamma+k\alpha_i}\cap T_i(U^+)$$ inductively by $$x_0=x$$, $$x_{k+1}=\frac1{[k+1]_{q_i}}(e_ix_k-q_i^{\langle\gamma,\alpha_i^\vee\rangle+2k}x_ke_i)$$. Then we see by induction on $$n$$ that   ei(n)x=∑k=0nqi(n−k)(⟨γ,αi∨⟩+k)xkei(n−k), (3.6) or equivalently,   xn=ei(n)x−∑k=0n−1qi(n−k)(⟨γ,αi∨⟩+k)xkei(n−k). (3.7) We obtain from (3.7) that $$x_n\in U^+_R$$ by induction on $$n$$. By $$T_i(U_R)=U_R$$ we have $$x_n\in U^+_R\cap T_i(U^+)=U^+_R\cap T_i(U^+_R)$$. It follows that $$e_i^{(n)} (U^+_R\cap T_i(U^+_R)) \subset \sum_{k=0}^n(U^+_R\cap T_i(U^+_R))e_i^{(k)}$$ by (3.6). ■ We set   ♯UR0=⨁γ∈QRkγ⊂UR0,♯UR≧0=♯UR0UR+,♯UR≦0=♯UR0UR−. Define a subring $${\tilde{{\mathbb{A}}}}$$ of $${\mathbb{F}}$$ by   A~=Z[q,q−1,(q−q−1)−1,[n]q−1∣n>0]=Z[q,q−1,(qn−1)−1∣n>0]. (3.8) Then the Drinfeld pairing induces a bilinear form   τA~:♯UA~≧0×♯UA~≦0→A~. For $$\gamma\in Q^+$$ we denote its restriction to $$U^+_{{\tilde{{\mathbb{A}}}},\gamma}\times U^-_{{\tilde{{\mathbb{A}}}},-\gamma}$$ by   τA~,γ:UA~,γ+×UA~,−γ−→A~. In the rest of this paper we fix a field $$K$$ and $$z\in K^\times$$ which is not a root of 1, and consider the Hopf algebra   Uz=K⊗A~UA~, (3.9) where $${\tilde{{\mathbb{A}}}}\to K$$ is given by $$q\mapsto z$$. We define subalgebras $$U_z^0$$, $$U_z^+$$, $$U_z^-$$, $$U_z^{\geqq0}$$, $$U_z^{\leqq0}$$ of $$U_z$$ by   Uz0=K⊗A~UA~0,Uz±=K⊗A~UA~±,Uz≧0=K⊗A~UA~≧0,Uz≦0=K⊗A~UA~≦0. For $$\gamma\in Q^+$$ we set $$U_{z,\pm\gamma}^{\pm} =K\otimes_{\tilde{{\mathbb{A}}}} U_{{\tilde{{\mathbb{A}}}},\pm\gamma}^{\pm}$$. Then we have   Uz0=⨁h∈hZKkh,Uz±=⨁γ∈Q+Uz,±γ±. By (3.1) we have   ∑γ∈Q+dim⁡Uz,−γ−e(−γ)=D−1. (3.10) Moreover, setting   Uz,γ={u∈Uz∣khukh−1=z⟨γ,h⟩u(h∈hZ)}(γ∈Q), we have $$U_{z,\pm\gamma}^{\pm}=U_z^\pm\cap U_{z,\gamma}$$ since $$z$$ is not a root of $$1$$. The multiplication of $$U_z$$ induces isomorphisms   Uz ≅Uz+⊗Uz0⊗Uz−≅Uz−⊗Uz0⊗Uz+, (3.11)  Uz≧0 ≅Uz+⊗Uz0≅Uz0⊗Uz+,Uz≦0≅Uz−⊗Uz0≅Uz0⊗Uz− (3.12) of $$K$$-modules. Here, $$\otimes$$ denotes $$\otimes_K$$. For a $$U_z$$-module $$V$$ and $$\lambda\in P$$ we set   Vλ={v∈V∣khv=z⟨λ,h⟩v(h∈hZ)}. We say that a $$U_z$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ and $$i\in I$$ there exists some $$N>0$$ such that $$e_i^{(n)}v=f_i^{(n)}v=0$$ for $$n\geqq N$$. For $$i\in I$$ and an integrable $$U_z$$-module $$V$$ define an operator $$T_i:V\to V$$ by   Tiv=∑−a+b−c=⟨λ,hi⟩(−1)bzi−ac+bei(a)fi(b)ei(c)v(v∈Vλ), where $$z_i=z^{(\alpha_i,\alpha_i)/2}$$. It is invertible, and satisfies $$T_iV_\lambda= V_{s_i\lambda}$$ for $$\lambda\in P$$. We denote by $$T_i:U_z\to U_z$$ the algebra automorphism of $$U_z$$ induced from $$T_i:U_{\tilde{{\mathbb{A}}}}\to U_{\tilde{{\mathbb{A}}}}$$. Then we have $$T_i(U_{z,\gamma})=U_{z,s_i\gamma}$$ for $$\gamma\in Q$$. Lemma 3.2. The multiplication of $$U_z$$ induces isomorphisms   Uz+≅(Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n)), (3.13)  Uz+≅(⨁n=0∞Kei(n))⊗(Uz+∩Ti−1(Uz+)), (3.14)  Uz−≅(Uz−∩Ti(Uz−))⊗(⨁n=0∞Kfi(n)), (3.15)  Uz−≅(⨁n=0∞Kfi(n))⊗(Uz−∩Ti−1(Uz−)). (3.16) □ Proof. We only show (3.13). By Lemma 3.1 we have   Uz+≅(K⊗A~(UA~+∩Ti(UA~+)))⊗(⨁n=0∞Kei(n)). By $$U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U_{\tilde{{\mathbb{A}}}}^+)=U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U^+)$$ the canonical map $$K\otimes_{\tilde{{\mathbb{A}}}}(U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U_{\tilde{{\mathbb{A}}}}^+)) \to U_z^+\cap T_i(U_z^+)$$ is injective. Hence we have a sequence of canonical maps   Uz+≅(K⊗A~(UA~+∩Ti(UA~+)))⊗(⨁n=0∞Kei(n))↪ (Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n))→Uz+. Therefore, it is sufficient to show that   (Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n))→Uz is injective. This follows by applying $$T_i$$ to $$U_z^+\otimes U_z^{\leqq0}\cong U_z$$. ■ We set   ♯Uz0=K⊗A~♯UA~0,♯Uz≧0=K⊗A~♯UA~≧0,♯Uz≦0=K⊗A~♯UA~≦0. They are Hopf subalgebras of $$U_z$$. The Drinfeld pairing induces a bilinear form   τz:♯Uz≧0×♯Uz≦0→K. For $$\gamma\in Q^+$$ we denote its restriction to $$U^+_{z,\gamma}\times U^-_{z,-\gamma}$$ by   τz,γ:Uz,γ+×Uz,−γ−→K. 4 The Modified Algebra Set   Jz+= {x∈Uz+∣τz(x,Uz−)={0}},Jz−= {y∈Uz−∣τz(Uz+,y)={0}}. For $$\gamma\in Q^+$$ we set   Jz,±γ±=Jz±∩Uz,±γ±. By (2.26) we have   Jz±=⨁γ∈Q+∖{0}Jz,±γ±. (4.1) Define a two-sided ideal $$J_z$$ of $$U_z$$ by   Jz=UzJz+Uz+UzJz−Uz. Proposition 4.1. (i) We have   Δ(Jz)⊂Uz⊗Jz+Jz⊗Uz,ε(Jz)={0},S(Jz)⊂Jz. (ii) Under the isomorphism $$U_z\cong U_z^+\otimes U_z^0\otimes U_z^-$$ (resp. $$U_z\cong U_z^-\otimes U_z^0\otimes U_z^+$$) induced by the multiplication of $$U_z$$ we have   Jz ≅Jz+⊗Uz0⊗Uz−+Uz+⊗Uz0⊗Jz−,(resp.Jz ≅Jz−⊗Uz0⊗Uz++Uz−⊗Uz0⊗Jz+). □ Proof. (i) It is sufficient to show   Δ(Jz+)⊂Jz+♯Uz0⊗Uz++♯Uz≧0⊗Jz+, (4.2)  Δ(Jz−)⊂Jz−⊗♯Uz≦0+♯Uz−⊗Jz−♯Uz0, (4.3)  ε(Jz±)= {0}, (4.4)  S(Jz±)⊂Jz±♯Uz0. (4.5) By (2.25) we have   Jz+♯Uz0={x∈♯Uz≧0∣τz(x,Uz−)={0}}. Hence in order to verify (4.2) it is sufficient to show   τz(Δ(Jz+),Uz−⊗Uz−)={0}. This follows from (2.20). The proof of (4.3) is similar. The assertions (4.4) and (4.5) follow from (4.1) and (2.28), respectively. (ii) It is sufficient to show   Jz±Uz±=Uz±Jz±=Jz±, (4.6)  Jz+Uz≦0=Uz≦0Jz+,Jz−Uz≧0=Uz≧0Jz−. (4.7) The assertion (4.6) follows from (2.20), (2.21), and (2.25). By (4.1) we have $$J_z^\pm U_z^0=U_z^0J_z^\pm$$. Hence in order to show (4.7) it is sufficient to show $$J_z^{+}{}^\sharp U_z^{\leqq0}={}^\sharp U_z^{\leqq0} J_z^{+}$$ and $$ J_z^{-}{}^\sharp U_z^{\geqq0}={}^\sharp U_z^{\geqq0} J_z^{-}$$. Let $$x\in J_z^{+}$$, $$y\in {}^\sharp U_z^{\leqq0}$$. By (4.2) we have   Δ2(x)∈♯Uz≧0⊗♯Uz≧0⊗Jz++♯Uz≧0⊗Jz+♯Uz0⊗Uz++Jz+♯Uz0⊗♯Uz≧0⊗Uz+. Hence we have $$xy\in {}^\sharp U_z^{\leqq0}J_z^{+}$$ and $$yx\in J_z^{+}{}^\sharp U_z^{\leqq0}$$ by (2.29) and (2.30). It follows that $$J_z^{+}{}^\sharp U_z^{\leqq0}={}^\sharp U_z^{\leqq0} J_z^{+}$$. The proof of $$J_z^{-}{}^\sharp U_z^{\geqq0}={}^\sharp U_z^{\geqq0} J_z^{-}$$ is similar. ■ By (2.35–2.37) we see easily the following. Lemma 4.2. For $$i\in I$$ we have   Jz− ≅(Jz−∩Ti(Uz−))⊗(⨁n=0∞Kfi(n)),Jz− ≅(⨁n=0∞Kfi(n))⊗(Jz−∩Ti−1(Uz−)). Moreover, we have   Ti−1(Jz−∩Ti(Uz−))=Jz−∩Ti−1(Uz−). □ We set   U¯z=Uz/Jz. (4.8) It is a Hopf algebra by Proposition 4.1. Denote by $$\overline{U}_z^0$$, $$\overline{U}_z^\pm$$, $$\overline{U}_z^{\geqq0}$$, $$\overline{U}_z^{\leqq0}$$, $${}^\sharp \overline{U}_z^{0}$$, $${}^\sharp \overline{U}_z^{\geqq0}$$, $${}^\sharp \overline{U}_z^{\leqq0}$$, $$\overline{U}_{z,\pm\gamma}^\pm$$;($$\gamma\in Q^+$$) the images of $${U}_z^0$$, $${U}_z^\pm$$, $${U}_z^{\geqq0}$$, $${U}_z^{\leqq0}$$, $${}^\sharp U_z^{0}$$, $${}^\sharp {U}_z^{\geqq0}$$, $${}^\sharp {U}_z^{\leqq0}$$, $${U}_{z,\pm\gamma}^\pm$$ under $$U_z\to\overline{U}_z$$, respectively. By the above argument we have   U¯z ≅U¯z+⊗U¯z0⊗U¯z−≅U¯z−⊗U¯z0⊗U¯z+,U¯z≧0 ≅U¯z+⊗U¯z0≅U¯z0⊗U¯z+,U¯z≦0≅U¯z−⊗U¯z0≅U¯z0⊗U¯z−,♯U¯z≧0 ≅U¯z+⊗♯U¯z0≅♯U¯z0⊗U¯z+,♯U¯z≦0≅U¯z−⊗♯U¯z0≅♯U¯z0⊗U¯z−,U¯z0 ≅Uz0=⨁h∈hZKkh,♯U¯z0≅♯Uz0=⨁γ∈QKkγ, and   U¯z±=⨁γ∈Q+U¯z,±γ±,U¯z,±γ±≅Uz,±γ±/Jz,±γ±. (4.9) By definition $$\tau_z$$ induces a bilinear form   τ¯z:♯U¯z≧0×♯U¯z≦0→K such that for any $$\gamma\in Q^+$$ its restriction   τ¯z,γ:U¯z,γ+×U¯z,−γ−→K is non-degenerate. For $$\lambda\in P$$ and a $$\overline{U}_z$$-module $$V$$ we set   Vλ={v∈V∣khv=z⟨λ,h⟩v(h∈hZ)}. We define a category $${\mathcal O}({\overline{U}}_z)$$ as follows. Its objects are $${\overline{U}}_z$$-modules $$V$$ which satisfy   V=⨁λ∈PVλ,dim⁡Vλ<∞(λ∈P), (4.10) and such that there exist finitely many $$\lambda_1,\ldots, \lambda_r\in P$$ such that   {λ∈P∣Vλ≠{0}}⊂⋃k=1r(λk−Q+). The morphisms are homomorphisms of $${\overline{U}}_z$$-modules. We say that a $${\overline{U}}_z$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ there exists $$N>0$$ such that for $$i\in I$$ and $$n\geqq N$$ we have $$e_i^{(n)}v=f_i^{(n)}v=0$$. We denote by $${\mathcal O}^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)$$ the full subcategory of $${\mathcal O}({\overline{U}}_z)$$ consisting of integrable $${\overline{U}}_z$$-modules belonging to $${\mathcal O}({\overline{U}}_z)$$. For each coset $$C=\mu+Q\in P/Q$$ we denote by $${\mathcal O}_C({\overline{U}}_z)$$ the full subcategory of $${\mathcal O}({\overline{U}}_z)$$ consisting of $$V\in {\mathcal O}_C({\overline{U}}_z)$$ such that $$V=\bigoplus_{\lambda\in C}V_\lambda$$. We also set $${\mathcal O}_C^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)={\mathcal O}_C({\overline{U}}_z)\cap{\mathcal O}^{\mathop{\rm int}\nolimits}({\overline{U}}_z)$$. Then we have   O(U¯z)=⨁C∈P/QOC(U¯z),Oint(U¯z)=⨁C∈P/QOCint(U¯z). (4.11) For $$\lambda\in P$$ we define $$M_z(\lambda)\in{\mathcal O}_{\lambda+Q}({\overline{U}}_z)$$ by   Mz(λ)=U¯z/(∑h∈hZU¯z(kh−z⟨λ,h⟩)+∑i∈IU¯zei), and for $$\lambda\in P^+$$ we define $$V_z(\lambda)\in{\mathcal O}_{\lambda+Q}^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)$$ by   Vz(λ)=U¯z/(∑h∈hZU¯z(kh−z⟨λ,h⟩)+∑i∈IU¯zei+∑i∈IU¯zfi(⟨λ,hi⟩+1)). Let $$\lambda\in P$$. A $${\overline{U}}_z$$-module $$V$$ is called a highest weight module with highest weight $$\lambda$$ if there exists $$v\in V_\lambda\setminus\{0\}$$ such that $$V={\overline{U}}_z v$$ and $$xv=\varepsilon(x)v$$;($$x\in {\overline{U}}_z^+$$). Then we have $$V\in{\mathcal O}_{\lambda+Q}({\overline{U}}_z)$$. A $${\overline{U}}_z$$-module is a highest weight module with highest weight $$\lambda$$ if and only if it is a non-zero quotient of $$M_z(\lambda)$$. If there exists an integrable highest weight module with highest weight $$\lambda$$, then we have $$\lambda\in P^+$$. For $$\lambda\in P^+$$ a $${\overline{U}}_z$$-module is an integrable highest weight module with highest weight $$\lambda$$ if and only if it is a non-zero quotient of $$V_z(\lambda)$$. For $$V\in{\mathcal O}({\overline{U}}_z)$$ we define its formal character by   ch⁡(V)=∑λ∈Pdim⁡Vλe(λ)∈E. We have   ch⁡(Mz(λ))=e(λ)D¯−1(λ∈P), where   D¯−1=∑γ∈Q+dim⁡U¯z,−γ−e(−γ)(λ∈P). For each coset $$C=\mu+Q\in P/Q$$ we fix a function $$f_C:C\to {\mathbb{Z}}$$ such that   fC(λ)−fC(λ−αi)=2⟨λ,ti⟩(λ∈C,i∈I). Remark 4.3. The function $$f_C$$ is unique up to addition of a constant function. If we extend $$(\;,\;):E\times E\to{\mathbb{Q}}$$ to a $$W$$-invariant symmetric bilinear form on $${\mathfrak{h}}^*$$, then $$f_C$$ is given by   fC(λ)=(λ+ρ,λ+ρ)+a(λ∈C) for some $$a\in {\mathbb{Q}}$$. □ For $$\gamma\in Q^+$$ let $$ \overline{C}_\gamma\in \overline{U}^+_{z,\gamma}\otimes \overline{U}^-_{z,-\gamma} $$ be the canonical element of the non-degenerate bilinear form $$\overline{\tau}_{z,\gamma}$$. Following Drinfeld we set   Ωγ=(m∘(S⊗1)∘P)(C¯γ)∈U¯z,−γ−U¯z0U¯z,γ+, where $$m:\overline{U}_z\otimes\overline{U}_z\to \overline{U}_z$$ and $$P:\overline{U}_z\otimes \overline{U}_z\to \overline{U}_z\otimes \overline{U}_z$$ are given by $$m(a,b)=ab$$, $$P(a\otimes b)=b\otimes a$$ (see [12, Section 3.2], [11, Section 6.1]). Let $$C\in P/Q$$. For $$V\in{\mathcal O}_C(\overline{U}_z)$$ we define a linear map   Ω:V→V (4.12) by   Ω(v)=zfC(λ)∑γ∈Q+Ωγv(v∈Vλ). This operator is called the quantum Casimir operator. As in [12, Section 3.2] we have the following. Proposition 4.4. Let $$C\in P/Q$$. For $$\lambda\in C$$ the operator $$\Omega$$ acts on $$M_z(\lambda)$$ as $$z^{f_C(\lambda)}\mathop{\rm id}\nolimits$$. □ Since $$z$$ is not a root of 1, we have   zfC(λ)=zfC(μ)⟹fC(λ)=fC(μ). 5 Main Results For $$w\in W$$ and $$x=\sum_{\lambda\in P}c_\lambda e(\lambda)\in{\mathcal E}$$ we set   wx=∑λ∈Pcλe(wλ),w∘x=∑λ∈Pcλe(w∘λ). The elements $$wx$$, $$w\circ x$$ may not belong to $${\mathcal E}$$; however, we will only consider the case where $$wx, w\circ x\in {\mathcal E}$$. We denote by $${\mathop{\rm sgn}\nolimits}:W\to\{\pm1\}$$ the character given by $${\mathop{\rm sgn}\nolimits}(s_i)=-1$$ for $$i\in I$$. Proposition 5.1. For any $$w\in W$$ we have $$w\circ \overline{D}={\mathop{\rm sgn}\nolimits}(w)\overline{D}$$. □ Proof. We may assume that $$w=s_i$$ for $$i\in I$$. Define $$D_i, \overline{D}_i\in{\mathcal E}$$ by   D =(1−e(−αi))Di,D¯=(1−e(−αi))D¯i. Then we have $$D_i=\prod_{\alpha\in\Delta^+\setminus\{\alpha_i\}}(1-e(-\alpha))^{m_\alpha}$$. Moreover, by Lemma 3.2, Lemma 4.2, and (4.9) we have   Di−1 =∑γ∈Q+dim⁡(Uz,−γ−∩Ti(Uz−))e(−γ) =∑γ∈Q+dim⁡(Uz,−γ−∩Ti−1(Uz−))e(−γ),D¯i−1 =Di−1−∑γ∈Q+dim⁡(Jz,−γ−∩Ti(Uz−))e(−γ) =Di−1−∑γ∈Q+dim⁡(Jz,−γ−∩Ti−1(Uz−))e(−γ). By $$s_i\circ\overline{D}=-(1-e(-\alpha_i))s_i\overline{D}_i$$ we have only to show $$s_i\overline{D}_i=\overline{D}_i$$. By Lemma 4.2 we have   si(∑γ∈Q+dim⁡(Jz,−γ−∩Ti(Uz−))e(−γ)) =∑γ∈Q+dim⁡(Jz−∩Ti(Uz,−γ−))e(−γ) =∑γ∈Q+dim⁡(Jz,−γ−∩Ti−1(Uz−))e(−γ), and hence the assertion follows from $$s_iD_i=D_i$$. ■ Proposition 5.2. Let $$\lambda\in P^+$$. Assume that $$V$$ is an integrable highest weight $${\overline{U}}_z$$-module with highest weight $$\lambda$$. Then we have   ch⁡(V)=∑w∈Wsgn(w)ch⁡(Mz(w∘λ)). □ Proof. The proof is the same as the one for Lie algebras in [6, Theorem 10.4]. Set $$C=\lambda+Q\in P/Q$$. Similarly to [6, Proposition 9.8] we have   ch⁡(V)=∑μ∈λ−Q+,fC(μ)=fC(λ)cμch⁡(Mz(μ))(cμ∈Z,cλ=1). (5.1) Multiplying (5.1) by $$\overline{D}$$ we obtain   D¯ch⁡(V)=∑μ∈λ−Q+,fC(μ)=fC(λ)cμe(μ). Using the action of $$T_i\;(i\in I)$$ on $$V$$ we see that $$w\mathop{\rm ch}\nolimits(V)=\mathop{\rm ch}\nolimits(V)$$ for $$w\in W$$, and hence $$w\circ(\overline{D}\mathop{\rm ch}\nolimits(V))={\mathop{\rm sgn}\nolimits}(w)\overline{D}\mathop{\rm ch}\nolimits(V)$$ for any $$w\in W$$. It follows that   cμ=sgn(w)cw∘μ(μ∈λ−Q+,w∈W). (5.2) Assume that $$\mu\in\lambda-Q^+$$ satisfies $$c_\mu\ne0$$. By (5.2) $$W\circ\mu\subset\lambda-Q^+$$, and hence we can take $$\mu'\in W\circ\mu$$ such that $${\mathop{\rm ht}\nolimits}(\lambda-\mu')$$ is minimal, where $${\mathop{\rm ht}\nolimits}(\sum_im_i\alpha_i)=\sum_im_i$$. Then we have $$\langle\mu',h_i\rangle\geqq0$$ for any $$i\in I$$ by $$s_i\circ\mu'= \mu'-(\langle\mu',h_i\rangle+1)\alpha_i$$ and (5.2). Namely, we have $$\mu'\in P^+$$. Then by [6, Lemma 10.3] we obtain $$\mu'=\lambda$$. □ Remark 5.3. Heckenberger pointed out to me that Proposition 5.2 also follows from the existence of the BGG resolution of integrable highest weight modules of quantized enveloping algebras given in [5] □ Recall that any integrable highest weight module $$V$$ with highest weight $$\lambda$$ is a quotient of $$V_z(\lambda)$$. Proposition 5.2 tells us that its character $$\mathop{\rm ch}\nolimits(V)$$ only depends on $$\lambda$$. It follows that any integrable highest weight module with highest weight $$\lambda$$ is isomorphic to $$V_z(\lambda)$$. Consider the case $$\lambda=0$$. Since $$V_z(0)$$ is the trivial one-dimensional module, we obtain the identity   1=(∑w∈Wsgn(w)e(w∘0))(∑γ∈Q+dim⁡U¯z,−γ−e(−γ)) in $${\mathcal E}$$ by Proposition 5.2. On the other hand by the corresponding result for the Kac–Moody Lie algebra we have   1=(∑w∈Wsgn(w)e(w∘0))(∑γ∈Q+dim⁡Uz,−γ−e(−γ)). It follows that $$U^-_{z,-\gamma}\cong{\overline{U}}^-_{z,-\gamma}$$ for any $$\gamma \in Q^+$$. By $$\dim U^-_{z,-\gamma}=\dim U^+_{z,\gamma}$$ and the non-degeneracy of $$\overline{\tau}_{z,\gamma}$$ we also have $$U^+_{z,\gamma}\cong{\overline{U}}^+_{z,\gamma}$$ for any $$\gamma\in Q^+$$. We have obtained the following results. Theorem 5.4 The Drinfeld pairing   τz,γ:Uz,γ+×Uz,−γ−→K is non-degenerate for any $$\gamma\in Q^+$$. □ Theorem 5.5 Let $$\lambda\in P^+$$. Assume that $$V$$ is an integrable highest weight $$U_z$$-module with highest weight $$\lambda$$. Then we have   ch⁡(V)=D−1∑w∈Wsgn(w)e(w∘λ). □ By Theorem 5.4 we can define the quantum Casimir operator $$\Omega$$ for $$U_z$$. As in [11, Section 6.2] we have the following. Theorem 5.6 Any object of $${\mathcal O}^{{\mathop{\rm int}\nolimits}}(U_z)$$ is a direct sum of $$V_z(\lambda)$$’s for $$\lambda\in P^+$$. □ By Theorem 5.4 we have the following. Theorem 5.7 Let $$\gamma\in Q^+$$. Take bases $$\{x_r\}$$ and $$\{y_s\}$$ of $$U^+_{{\mathbb{A}},\gamma}$$ and $$U^-_{{\mathbb{A}},-\gamma}$$, respectively, and set $$f_\gamma=\det(\tau_{{\tilde{{\mathbb{A}}}},\gamma}(x_r,y_s))_{r,s}$$. Then we have $$f_\gamma\in{\tilde{{\mathbb{A}}}}^\times$$. Namely, we have   fγ=±qaf1±1⋯fN±1, where $$a\in{\mathbb{Z}}$$, and $$f_1,\dots, f_N\in{\mathbb{Z}}[q]$$ are cyclotomic polynomials. □ Proof. We can write $$f_\gamma=mgh$$, where $$m\in{\mathbb{Z}}_{>0}$$, $$g\in {\mathbb{Z}}[q]$$ is a primitive polynomial with $$g(0)>0$$ whose irreducible factor is not cyclotomic, and $$h\in{\tilde{{\mathbb{A}}}}^\times$$. For any field $$K$$ and $$z\in K^\times$$ which is not a root of 1, the specialization of $$f_\gamma$$ with respect to the ring homomorphism $${\tilde{{\mathbb{A}}}}\to K\;(q\mapsto z)$$ is non-zero by Theorem 5.4. Hence we see easily that $$m=1$$ and $$g=1$$. ■ In the finite case Theorem 5.7 is well-known [8, 9, 11]. In the affine case this is a consequence of [3, 4], where $$\det(\tau_{{\tilde{{\mathbb{A}}}},\gamma}(x_r,y_s))_{r,s}$$ is determined explicitly by a case-by-case calculation. Funding This work was supported by Grants-in-Aid for Scientific Research (C) 15K04790 from Japan Society for the Promotion of Science. Acknowledgement The first draft of this paper contained only results when $$K$$ is of characteristic zero. 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[15] Tsuchioka S. “Graded Cartan determinants of the symmetric groups.” Transactions of the American Mathematical Society  366, no. 4 ( 2014): 2019– 40. Google Scholar CrossRef Search ADS   Communicated by Prof. Masaki Kashiwara © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Mathematics Research Notices Oxford University Press

Characters of Integrable Highest Weight Modules over a Quantum Group

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Abstract

Abstract We show that the Weyl–Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac–Moody Lie algebra, when the parameter $$q$$ is not a root of unity. 1 Introduction It is well-known that the character of an integrable highest weight module over a symmetrizable Kac–Moody algebra $${\mathfrak{g}}$$ is given by the Weyl–Kac character formula [6]. In this paper we consider the corresponding problem for a quantized enveloping algebra [7]. For a field $$K$$ and $$z\in K^\times$$ which is not a root of 1, we denote by $$U_{K,z}({\mathfrak{g}})$$ the quantized enveloping algebra of $${\mathfrak{g}}$$ over $$K$$ at $$q=z$$, namely the specialization of Lusztig’s $${\mathbb{Z}}[q,q^{-1}]$$-form via $$q\mapsto z$$. It is already known that the Weyl–Kac type character formula holds for $$U_{K,z}({\mathfrak{g}})$$ in some cases. When $$K$$ is of characteristic $$0$$ and $$z$$ is transcendental, this is due to Lusztig [10]. When $${\mathfrak{g}}$$ is finite-dimensional, this is shown in [1]. When $${\mathfrak{g}}$$ is affine, this is known in certain specific cases [2, 15]. We first point out that the problem is closely related to the non-degeneracy of the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$. In fact, assume we could show that the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ is non-degenerate. Then we can define the quantum Casimir operator. It allows us to apply Kac’s argument for Lie algebras in [6] to $$U_{K,z}({\mathfrak{g}})$$, and we obtain the Weyl–Kac type character formula for integrable highest weight modules over $$U_{K,z}({\mathfrak{g}})$$. In particular, we can deduce the Weyl–Kac type character formula in the affine case from the case-by-case calculation of the Drinfeld pairing due to Damiani [3, 4]. The aim of this paper is to give a simple unified proof of the non-degeneracy of the Drinfeld pairing and the Weyl–Kac type character formula for $$U_{K,z}({\mathfrak{g}})$$, where $${\mathfrak{g}}$$ is a symmetrizable Kac–Moody algebra, $$K$$ is a field not necessarily of characteristic zero, and $$z\in K^\times$$ is not a root of 1. Our argument is as follows. We consider the (possibly) modified algebra $${\overline{U}}_{K,z}({\mathfrak{g}})$$, which is the quotient of $$U_{K,z}({\mathfrak{g}})$$ by the ideal generated by the radical of the Drinfeld pairing. Then the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ induces a non-degenerate pairing for $${\overline{U}}_{K,z}({\mathfrak{g}})$$, by which we can define the quantum Casimir operator for $${\overline{U}}_{K,z}({\mathfrak{g}})$$. It allows us to apply Kac’s argument for Lie algebras to $${\overline{U}}_{K,z}({\mathfrak{g}})$$, and we obtain the Weyl–Kac type character formula for $${\overline{U}}_{K,z}({\mathfrak{g}})$$ with modified denominator. In the special case where the highest weight is zero, this gives a formula for the modified denominator. Comparing this with the ordinary denominator formula for Lie algebras, we conclude that the modified denominator coincides with the original denominator for the Lie algebra $${\mathfrak{g}}$$. It implies that the Drinfeld pairing for $$U_{K,z}({\mathfrak{g}})$$ was already non-degenerate. This is the outline of our argument. In applying Kac’s argument to the modified algebra, we need to show that the modified denominator is skew invariant with respect to a twisted action of the Weyl group. This is accomplished using certain standard properties of the Drinfeld pairing. 2 Quantized Enveloping Algebras Let $${\mathfrak{h}}$$ be a finite-dimensional vector space over $${\mathbb{Q}}$$, and let $$\{h_i\}_{i\in I}$$ and $$\{\alpha_i\}_{i\in I}$$ be linearly independent subsets of $${\mathfrak{h}}$$ and $${\mathfrak{h}}^*$$, respectively, such that $$(\langle \alpha_j,h_i\rangle)_{i, j\in I}$$ is a symmetrizable generalized Cartan matrix. We denote by $$W$$ the associated Weyl group. It is a subgroup of $$GL({\mathfrak{h}})$$ generated by the involutions $$s_i$$ ($$i\in I$$) defined by $$s_i(h)=h-\langle\alpha_i,h\rangle h_i$$ for $$h\in {\mathfrak{h}}$$. The contragredient action of $$W$$ on $${\mathfrak{h}}^*$$ is given by $$s_i(\lambda)=\lambda-\langle\lambda,h_i\rangle \alpha_i$$ for $$i\in I$$, $$\lambda\in {\mathfrak{h}}^*$$. Set   E=∑i∈IQαi,Q=∑i∈IZαi,Q+=∑i∈IZ≧0αi. We can take a symmetric $$W$$-invariant bilinear form $$ (\;,\;): E\times E\to{\mathbb{Q}} $$ such that   (αi,αi)2∈Z>0(i∈I). (2.1) For $$\lambda\in E$$ and $$i\in I$$ we obtain from $$(\lambda,\alpha_i)=(s_i\lambda,s_i\alpha_i)$$ that   ⟨λ,hi⟩=2(λ,αi)(αi,αi). (2.2) In particular we have   (αi,αj)=⟨αj,hi⟩(αi,αi)2∈Z, and hence $$(Q,Q)\subset{\mathbb{Z}}$$. For $$i\in I$$ set $$t_i=\frac{(\alpha_i,\alpha_i)}2h_i$$, and for $$\gamma=\sum_in_i\alpha_i\in Q$$ set $$t_\gamma=\sum_in_it_i$$. By (2.2) we have $$(\lambda,\gamma)=\langle\lambda,t_\gamma\rangle$$ for $$\lambda\in E$$, $$\gamma\in Q$$. We fix a $${\mathbb{Z}}$$-form $${\mathfrak{h}}_{\mathbb{Z}}$$ of $${\mathfrak{h}}$$ such that   ⟨αi,hZ⟩⊂Z,ti∈hZ(i∈I). (2.3) We set   P={λ∈h∗∣⟨λ,hZ⟩⊂Z},P+={λ∈P∣⟨λ,hi⟩∈Z≧0}. We fix $$\rho\in{\mathfrak{h}}^*$$ such that $$\langle\rho, h_i\rangle=1$$ for any $$i\in I$$, and define a twisted action of $$W$$ on $${\mathfrak{h}}^*$$ by   w∘λ=w(λ+ρ)−ρ(w∈W,λ∈h∗). This action does not depend on the choice of $$\rho$$, and we have $$w\circ P=P$$ for any $$w\in W$$. Denote by $${\mathcal E}$$ the set of formal sums $$\sum_{\lambda\in P}c_\lambda e(\lambda)$$; ($$c_\lambda\in {\mathbb{Z}}$$) such that there exist finitely many $$\lambda_1,\dots, \lambda_r\in P$$ such that   {λ∈P∣cλ≠0}⊂⋃k=1r(λk−Q+). Note that $${\mathcal E}$$ is naturally a commutative ring by the multiplication $$e(\lambda)e(\mu)=e(\lambda+\mu)$$. Denote by $$\Delta^+$$ the set of positive roots for the Kac–Moody Lie algebra $${\mathfrak{g}}$$ associated with the generalized Cartan matrix $$(\langle\alpha_j, h_i\rangle)_{i,j\in I}$$. For $$\alpha\in\Delta^+$$ let $$m_\alpha$$ be the dimension of the root space of $${\mathfrak{g}}$$ with weight $$\alpha$$. We define an invertible element $$D$$ of $${\mathcal E}$$ by   D=∏α∈Δ+(1−e(−α))mα. For $$n\in{\mathbb{Z}}_{\geqq0}$$ set   [n]x=xn−x−nx−x−1∈Z[x,x−1],[n]!x=[n]x[n−1]x⋯[1]x∈Z[x,x−1]. We denote by $${\mathbb{F}}={\mathbb{Q}}(q)$$ the field of rational functions in the variable $$q$$ with coefficients in $${\mathbb{Q}}$$. The quantized enveloping algebra $$U$$ associated with $${\mathfrak{h}}$$, $$\{h_i\}_{i\in I}$$, $$\{\alpha_i\}_{i\in I}$$, $${\mathfrak{h}}_{\mathbb{Z}}$$, $$(\;,\;)$$ is the associative algebra over $${\mathbb{F}}$$ generated by the elements $$k_h$$, $$e_i$$, $$f_i$$ ($$h\in{\mathfrak{h}}_{\mathbb{Z}}$$, $$i\in I$$) satisfying the relations   k0=1,khkh′=kh+h′ (h,h′∈hZ), (2.4)  kheik−h=qi⟨αi,h⟩ei (h∈hZ,i∈I), (2.5)  khfik−h=qi−⟨αi,h⟩fi (h∈hZ,i∈I), (2.6)  eifj−fjei=δijki−ki−1qi−qi−1 (i,j∈I), (2.7)   ∑r+s=1−⟨αj,hi⟩(−1)rei(r)ejei(s)=0 (i,j∈I,i≠j), (2.8)   ∑r+s=1−⟨αj,hi⟩(−1)rfi(r)fjfi(s)=0 (i,j∈I,i≠j), (2.9) where $$k_i=k_{t_i}$$, $$q_i=q^{(\alpha_i,\alpha_i)/2}$$ for $$i\in I$$, and $$e_i^{(r)}=\frac1{[r]!_{q_i}}e_i^r$$, $$f_i^{(r)}=\frac1{[r]!_{q_i}}f_i^r$$ for $$i\in I$$, $$r\in{\mathbb{Z}}_{\geqq0}$$. For $$\gamma\in Q$$ we set $$k_\gamma=k_{t_\gamma}$$. We have a Hopf algebra structure of $$U$$ given by   Δ(kh)=kh⊗kh,Δ(ei)=ei⊗1+ki⊗ei,Δ(fi)=fi⊗ki−1+1⊗fi, (2.10)  ε(kh)=1,ε(ei)=ε(fi)=0, (2.11)  S(kh)=kh−1,S(ei)=−ki−1ei,S(fi)=−fiki (2.12) for $$h\in{\mathfrak{h}}_{\mathbb{Z}}, i\in I$$. We will sometimes use Sweedler’s notation for the coproduct   Δ(u)=∑(u)u(0)⊗u(1)(u∈U), and the iterated coproduct   Δm(u)=∑(u)mu(0)⊗⋯⊗u(m)(u∈U). We define $${\mathbb{F}}$$-subalgebras $$U^0$$, $$U^+$$, $$U^-$$, $$U^{\geqq0}$$, $$U^{\leqq0}$$ of $$U$$ by   U0=⟨kh∣h∈hZ⟩,U+=⟨ei∣i∈I⟩,U−=⟨fi∣i∈I⟩,U≧0=⟨kh,ei∣h∈hZ,i∈I⟩,U≦0=⟨kh,fi∣h∈hZ,i∈I⟩. For $$\gamma\in Q$$ set   Uγ={u∈U∣khukh−1=q⟨γ,h⟩u(h∈hZ)},Uγ±=Uγ∩U±. Then we have   U0=⨁h∈hZFkh,U±=⨁γ∈Q+U±γ±. It is known that the multiplication of $$U$$ induces isomorphisms   U≅U+⊗U0⊗U−≅U−⊗U0⊗U+,U≧0≅U+⊗U0≅U0⊗U+,U≦0≅U−⊗U0≅U0⊗U− of vector spaces. It is also known that   ∑γ∈Q+dim⁡U−γ−e(−γ)=D−1. (2.13) For a $$U$$-module $$V$$ and $$\lambda\in P$$ we set   Vλ={v∈V∣khv=q⟨λ,h⟩v(h∈hZ)}. We say that a $$U$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ and $$i\in I$$ there exists some $$N>0$$ such that $$e_i^{(n)}v=f_i^{(n)}v=0$$ for $$n\geqq N$$. For $$i\in I$$ and an integrable $$U$$-module $$V$$ define an operator $$T_i:V\to V$$ by   Tiv=∑−a+b−c=⟨λ,hi⟩(−1)bqi−ac+bei(a)fi(b)ei(c)v(v∈Vλ). It is invertible and satisfies $$T_iV_\lambda= V_{s_i\lambda}$$ for $$\lambda\in P$$. There exists a unique algebra automorphism $$T_i:U\to U$$ such that for any integrable $$U$$-module $$V$$ we have $$ T_iuv=T_i(u)T_iv$$;($$u\in U, v\in V$$). Then we have $$T_i(U_\gamma)=U_{s_i\gamma}$$ for $$\gamma\in Q$$. The action of $$T_i$$ on $$U$$ is given by   Ti(kh) =ksih,Ti(ei)=−fiki,Ti(fi)=−ki−1ei(h∈hZ),Ti(ej) =∑r+s=−⟨αj,hi⟩(−1)rqi−rei(s)ejei(r)(j∈I,i≠j),Ti(fj) =∑r+s=−⟨αj,hi⟩(−1)rqirfi(r)fjfi(s)(j∈I,i≠j) [11, Section 37.1]. The multiplication of $$U$$ induces   U+≅(U+∩Ti(U+))⊗F[ei]≅F[ei]⊗(U+∩Ti−1(U+)), (2.14)  U−≅(U−∩Ti(U−))⊗F[fi]≅F[fi]⊗(U−∩Ti−1(U−)) (2.15) [11, Lemma 38.1.2]. Moreover,   Δ(U+∩Ti(U+))⊂U≧0⊗(U+∩Ti(U+)), (2.16)  Δ(U+∩Ti−1(U+))⊂U0(U+∩Ti−1(U+))⊗U+, (2.17)  Δ(U−∩Ti(U−))⊂(U−∩Ti(U−))⊗U≦0, (2.18)  Δ(U−∩Ti−1(U−))⊂U−⊗U0(U−∩Ti−1(U−)) (2.19) [13, Lemma 2.8]. Set   ♯U0=⨁γ∈QFkγ⊂U0,♯U≧0=♯U0U+,♯U≦0=♯U0U−. They are Hopf subalgebras of $$U$$. The Drinfeld pairing is the bilinear form   τ:♯U≧0×♯U≦0→F characterized by the following properties:   τ(x,y1y2)=(τ⊗τ)(Δ(x),y1⊗y2) (x∈♯U≧0,y1,y2∈♯U≦0), (2.20)  τ(x1x2,y)=(τ⊗τ)(x2⊗x1,Δ(y)) (x1,x2∈♯U≧0,y∈♯U≦0), (2.21)  τ(kγ,kδ)=q−(γ,δ) (γ,δ∈Q), (2.22)  τ(ei,fj)=−δij(qi−qi−1)−1 (i,j∈I), (2.23)  τ(ei,kγ)=τ(kγ,fi)=0 (i∈I,γ∈Q). (2.24) It satisfies the following properties:   τ(xkγ,ykδ)=τ(x,y)q−(γ,δ) (x∈U+,y∈U−,γ,δ∈Q), (2.25)  τ(Uγ+,U−δ−)={0} (γ,δ∈Q+,γ≠δ), (2.26)  τ|Uγ+×U−γ−is non-degenerate (γ∈Q+), (2.27)  τ(Sx,Sy)=τ(x,y) (x∈♯U≧0,y∈♯U≦0). (2.28) Moreover, for $$x\in{}^\sharp{U}^{\geqq0}$$, $$y\in {}^\sharp{U}^{\leqq0}$$ we have   xy=∑(x)2,(y)2τ(x(0),y(0))τ(x(2),Sy(2))y(1)x(1), (2.29)  yx=∑(x)2,(y)2τ(Sx(0),y(0))τ(x(2),y(2))x(1)y(1) (2.30) [12, Lemma 2.1.2]. For $$i\in I$$ we define linear maps   ri,±:U±→U±,ri,±′:U±→U± by   Δ(x)∈ri,+(x)ki⊗ei+∑δ∈Q+∖{αi}U≧0⊗Uδ+ (x∈U+),Δ(x)∈eikγ−αi⊗ri,+′(x)+∑δ∈Q+∖{αi}Uδ+U0⊗U+ (x∈Uγ+),Δ(y)∈ri,−(y)⊗fik−γ+αi+∑δ∈Q+∖{αi}U−⊗U−δ−U0 (y∈U−γ−),Δ(y)∈fi⊗ri,−′(y)ki−1+∑δ∈Q+∖{αi}U−δ−⊗U≦0 (y∈U−). We have   U+∩Ti(U+)= {u∈U+∣τ(u,U−fi)={0}}={u∈U+∣ri,+(u)=0}, (2.31)  U+∩Ti−1(U+)= {u∈U+∣τ(u,fiU−)={0}}={u∈U+∣ri,+′(u)=0}, (2.32)  U−∩Ti(U−)= {u∈U−∣τ(U+ei,u)={0}}={u∈U−∣ri,−′(u)=0}, (2.33)  U−∩Ti−1(U−)= {u∈U−∣τ(eiU+,u)={0}}={u∈U−∣ri,−(u)=0} (2.34) [11, Proposition 38.1.6]. By (2.16–2.19) and (2.31–2.34) we easily obtain   τ(xeim,yfin)=δmnτ(x,y)qin(n−1)/2(qi−1−qi)n[n]!qi (x∈U+∩Ti(U+),y∈U−∩Ti(U−)), (2.35)  τ(eimx′,finy′)=δmnτ(x′,y′)qin(n−1)/2(qi−1−qi)n[n]!qi (x′∈U+∩Ti−1(U+),y′∈U−∩Ti−1(U−)). (2.36) We have also   τ(x,y)=τ(Ti−1(x),Ti−1(y))(x∈U+∩Ti(U+),y∈U−∩Ti(U−)) (2.37) [11, Proposition 38.2.1], [14, Theorem 5.1]. 3 Specialization Let $$R$$ be a subring of $${\mathbb{F}}={\mathbb{Q}}(q)$$ containing $${\mathbb{A}}={\mathbb{Z}}[q,q^{-1}]$$. We denote by $$U_R$$ the $$R$$-subalgebra of $$U$$ generated by $$k_h$$, $$e_i^{(n)}$$, $$f_i^{(n)}$$ ($$h\in{\mathfrak{h}}_{\mathbb{Z}}, i\in I, n\geqq0$$). It is a Hopf algebra over $$R$$. We define subalgebras $$U_R^0$$, $$U_R^+$$, $$U_R^-$$, $$U_R^{\geqq0}$$, $$U_R^{\leqq0}$$ of $$U_R$$ by   UR0=U0∩UR,UR±=U±∩UR,UR≧0=U≧0∩UR,UR≦0=U≦0∩UR. Setting $$ U_{R,\pm\gamma}^\pm=U_{\pm\gamma}^\pm\cap U_R$$ for $$\gamma\in Q^+ $$ we have   UR±=⨁γ∈Q+UR,±γ±. It is known that $$U^\pm_{R,\pm\gamma}$$ is a free $$R$$-module of rank $$\dim U^\pm_{\pm\gamma}$$ [11, Section 14.2]. Hence we have   ∑γ∈Q+rankR(UR,−γ−)e(−γ)=D−1 (3.1) by (2.13). The multiplication of $$U_R$$ induces isomorphisms   UR≅UR+⊗UR0⊗UR−≅UR−⊗UR0⊗UR+,UR≧0≅UR+⊗UR0≅UR0⊗UR+,UR≦0≅UR−⊗UR0≅UR0⊗UR− of $$R$$-modules. For $$i\in I$$ the algebra automorphisms $$T_i^{\pm1}:U\to U$$ preserve $$U_R$$. Lemma 3.1. The multiplication of $$U_R$$ induces isomorphisms   UR+≅(UR+∩Ti(UR+))⊗R(⨁n=0∞Rei(n)), (3.2)  UR+≅(⨁n=0∞Rei(n))⊗R(UR+∩Ti−1(UR+)), (3.3)  UR−≅(UR−∩Ti(UR−))⊗R(⨁n=0∞Rfi(n)), (3.4)  UR−≅(⨁n=0∞Rfi(n))⊗R(UR−∩Ti−1(UR−)). (3.5) □ Proof. We only show (3.2). The injectivity of the canonical homomorphism   (UR+∩Ti(UR+))⊗R(⨁n=0∞Rei(n))→UR+ is clear. To show the surjectivity it is sufficient to verify that its image is stable under the left multiplication by $$e_j^{(n)}$$ for any $$j\in I$$ and $$n\geqq0$$. If $$j\ne i$$, this is clear since $$e_j^{(n)}\in U_R^+\cap T_i(U_R^+)$$. Consider the case $$j=i$$. By (2.31) and the general formula   ri,+(xx′)=qi⟨γ′,αi∨⟩ri,+(x)x′+xri,+(x′)(x∈U+,x′∈Uγ′+) we easily obtain   x∈Uγ+∩Ti(U+)⟹eix−qi⟨γ,αi∨⟩xei∈Uγ+αi+∩Ti(U+). Now let $$x\in U^+_{R,\gamma}\cap T_i(U^+_{R})$$. Define $$x_k\in U^+_{\gamma+k\alpha_i}\cap T_i(U^+)$$ inductively by $$x_0=x$$, $$x_{k+1}=\frac1{[k+1]_{q_i}}(e_ix_k-q_i^{\langle\gamma,\alpha_i^\vee\rangle+2k}x_ke_i)$$. Then we see by induction on $$n$$ that   ei(n)x=∑k=0nqi(n−k)(⟨γ,αi∨⟩+k)xkei(n−k), (3.6) or equivalently,   xn=ei(n)x−∑k=0n−1qi(n−k)(⟨γ,αi∨⟩+k)xkei(n−k). (3.7) We obtain from (3.7) that $$x_n\in U^+_R$$ by induction on $$n$$. By $$T_i(U_R)=U_R$$ we have $$x_n\in U^+_R\cap T_i(U^+)=U^+_R\cap T_i(U^+_R)$$. It follows that $$e_i^{(n)} (U^+_R\cap T_i(U^+_R)) \subset \sum_{k=0}^n(U^+_R\cap T_i(U^+_R))e_i^{(k)}$$ by (3.6). ■ We set   ♯UR0=⨁γ∈QRkγ⊂UR0,♯UR≧0=♯UR0UR+,♯UR≦0=♯UR0UR−. Define a subring $${\tilde{{\mathbb{A}}}}$$ of $${\mathbb{F}}$$ by   A~=Z[q,q−1,(q−q−1)−1,[n]q−1∣n>0]=Z[q,q−1,(qn−1)−1∣n>0]. (3.8) Then the Drinfeld pairing induces a bilinear form   τA~:♯UA~≧0×♯UA~≦0→A~. For $$\gamma\in Q^+$$ we denote its restriction to $$U^+_{{\tilde{{\mathbb{A}}}},\gamma}\times U^-_{{\tilde{{\mathbb{A}}}},-\gamma}$$ by   τA~,γ:UA~,γ+×UA~,−γ−→A~. In the rest of this paper we fix a field $$K$$ and $$z\in K^\times$$ which is not a root of 1, and consider the Hopf algebra   Uz=K⊗A~UA~, (3.9) where $${\tilde{{\mathbb{A}}}}\to K$$ is given by $$q\mapsto z$$. We define subalgebras $$U_z^0$$, $$U_z^+$$, $$U_z^-$$, $$U_z^{\geqq0}$$, $$U_z^{\leqq0}$$ of $$U_z$$ by   Uz0=K⊗A~UA~0,Uz±=K⊗A~UA~±,Uz≧0=K⊗A~UA~≧0,Uz≦0=K⊗A~UA~≦0. For $$\gamma\in Q^+$$ we set $$U_{z,\pm\gamma}^{\pm} =K\otimes_{\tilde{{\mathbb{A}}}} U_{{\tilde{{\mathbb{A}}}},\pm\gamma}^{\pm}$$. Then we have   Uz0=⨁h∈hZKkh,Uz±=⨁γ∈Q+Uz,±γ±. By (3.1) we have   ∑γ∈Q+dim⁡Uz,−γ−e(−γ)=D−1. (3.10) Moreover, setting   Uz,γ={u∈Uz∣khukh−1=z⟨γ,h⟩u(h∈hZ)}(γ∈Q), we have $$U_{z,\pm\gamma}^{\pm}=U_z^\pm\cap U_{z,\gamma}$$ since $$z$$ is not a root of $$1$$. The multiplication of $$U_z$$ induces isomorphisms   Uz ≅Uz+⊗Uz0⊗Uz−≅Uz−⊗Uz0⊗Uz+, (3.11)  Uz≧0 ≅Uz+⊗Uz0≅Uz0⊗Uz+,Uz≦0≅Uz−⊗Uz0≅Uz0⊗Uz− (3.12) of $$K$$-modules. Here, $$\otimes$$ denotes $$\otimes_K$$. For a $$U_z$$-module $$V$$ and $$\lambda\in P$$ we set   Vλ={v∈V∣khv=z⟨λ,h⟩v(h∈hZ)}. We say that a $$U_z$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ and $$i\in I$$ there exists some $$N>0$$ such that $$e_i^{(n)}v=f_i^{(n)}v=0$$ for $$n\geqq N$$. For $$i\in I$$ and an integrable $$U_z$$-module $$V$$ define an operator $$T_i:V\to V$$ by   Tiv=∑−a+b−c=⟨λ,hi⟩(−1)bzi−ac+bei(a)fi(b)ei(c)v(v∈Vλ), where $$z_i=z^{(\alpha_i,\alpha_i)/2}$$. It is invertible, and satisfies $$T_iV_\lambda= V_{s_i\lambda}$$ for $$\lambda\in P$$. We denote by $$T_i:U_z\to U_z$$ the algebra automorphism of $$U_z$$ induced from $$T_i:U_{\tilde{{\mathbb{A}}}}\to U_{\tilde{{\mathbb{A}}}}$$. Then we have $$T_i(U_{z,\gamma})=U_{z,s_i\gamma}$$ for $$\gamma\in Q$$. Lemma 3.2. The multiplication of $$U_z$$ induces isomorphisms   Uz+≅(Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n)), (3.13)  Uz+≅(⨁n=0∞Kei(n))⊗(Uz+∩Ti−1(Uz+)), (3.14)  Uz−≅(Uz−∩Ti(Uz−))⊗(⨁n=0∞Kfi(n)), (3.15)  Uz−≅(⨁n=0∞Kfi(n))⊗(Uz−∩Ti−1(Uz−)). (3.16) □ Proof. We only show (3.13). By Lemma 3.1 we have   Uz+≅(K⊗A~(UA~+∩Ti(UA~+)))⊗(⨁n=0∞Kei(n)). By $$U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U_{\tilde{{\mathbb{A}}}}^+)=U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U^+)$$ the canonical map $$K\otimes_{\tilde{{\mathbb{A}}}}(U_{\tilde{{\mathbb{A}}}}^+\cap T_i(U_{\tilde{{\mathbb{A}}}}^+)) \to U_z^+\cap T_i(U_z^+)$$ is injective. Hence we have a sequence of canonical maps   Uz+≅(K⊗A~(UA~+∩Ti(UA~+)))⊗(⨁n=0∞Kei(n))↪ (Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n))→Uz+. Therefore, it is sufficient to show that   (Uz+∩Ti(Uz+))⊗(⨁n=0∞Kei(n))→Uz is injective. This follows by applying $$T_i$$ to $$U_z^+\otimes U_z^{\leqq0}\cong U_z$$. ■ We set   ♯Uz0=K⊗A~♯UA~0,♯Uz≧0=K⊗A~♯UA~≧0,♯Uz≦0=K⊗A~♯UA~≦0. They are Hopf subalgebras of $$U_z$$. The Drinfeld pairing induces a bilinear form   τz:♯Uz≧0×♯Uz≦0→K. For $$\gamma\in Q^+$$ we denote its restriction to $$U^+_{z,\gamma}\times U^-_{z,-\gamma}$$ by   τz,γ:Uz,γ+×Uz,−γ−→K. 4 The Modified Algebra Set   Jz+= {x∈Uz+∣τz(x,Uz−)={0}},Jz−= {y∈Uz−∣τz(Uz+,y)={0}}. For $$\gamma\in Q^+$$ we set   Jz,±γ±=Jz±∩Uz,±γ±. By (2.26) we have   Jz±=⨁γ∈Q+∖{0}Jz,±γ±. (4.1) Define a two-sided ideal $$J_z$$ of $$U_z$$ by   Jz=UzJz+Uz+UzJz−Uz. Proposition 4.1. (i) We have   Δ(Jz)⊂Uz⊗Jz+Jz⊗Uz,ε(Jz)={0},S(Jz)⊂Jz. (ii) Under the isomorphism $$U_z\cong U_z^+\otimes U_z^0\otimes U_z^-$$ (resp. $$U_z\cong U_z^-\otimes U_z^0\otimes U_z^+$$) induced by the multiplication of $$U_z$$ we have   Jz ≅Jz+⊗Uz0⊗Uz−+Uz+⊗Uz0⊗Jz−,(resp.Jz ≅Jz−⊗Uz0⊗Uz++Uz−⊗Uz0⊗Jz+). □ Proof. (i) It is sufficient to show   Δ(Jz+)⊂Jz+♯Uz0⊗Uz++♯Uz≧0⊗Jz+, (4.2)  Δ(Jz−)⊂Jz−⊗♯Uz≦0+♯Uz−⊗Jz−♯Uz0, (4.3)  ε(Jz±)= {0}, (4.4)  S(Jz±)⊂Jz±♯Uz0. (4.5) By (2.25) we have   Jz+♯Uz0={x∈♯Uz≧0∣τz(x,Uz−)={0}}. Hence in order to verify (4.2) it is sufficient to show   τz(Δ(Jz+),Uz−⊗Uz−)={0}. This follows from (2.20). The proof of (4.3) is similar. The assertions (4.4) and (4.5) follow from (4.1) and (2.28), respectively. (ii) It is sufficient to show   Jz±Uz±=Uz±Jz±=Jz±, (4.6)  Jz+Uz≦0=Uz≦0Jz+,Jz−Uz≧0=Uz≧0Jz−. (4.7) The assertion (4.6) follows from (2.20), (2.21), and (2.25). By (4.1) we have $$J_z^\pm U_z^0=U_z^0J_z^\pm$$. Hence in order to show (4.7) it is sufficient to show $$J_z^{+}{}^\sharp U_z^{\leqq0}={}^\sharp U_z^{\leqq0} J_z^{+}$$ and $$ J_z^{-}{}^\sharp U_z^{\geqq0}={}^\sharp U_z^{\geqq0} J_z^{-}$$. Let $$x\in J_z^{+}$$, $$y\in {}^\sharp U_z^{\leqq0}$$. By (4.2) we have   Δ2(x)∈♯Uz≧0⊗♯Uz≧0⊗Jz++♯Uz≧0⊗Jz+♯Uz0⊗Uz++Jz+♯Uz0⊗♯Uz≧0⊗Uz+. Hence we have $$xy\in {}^\sharp U_z^{\leqq0}J_z^{+}$$ and $$yx\in J_z^{+}{}^\sharp U_z^{\leqq0}$$ by (2.29) and (2.30). It follows that $$J_z^{+}{}^\sharp U_z^{\leqq0}={}^\sharp U_z^{\leqq0} J_z^{+}$$. The proof of $$J_z^{-}{}^\sharp U_z^{\geqq0}={}^\sharp U_z^{\geqq0} J_z^{-}$$ is similar. ■ By (2.35–2.37) we see easily the following. Lemma 4.2. For $$i\in I$$ we have   Jz− ≅(Jz−∩Ti(Uz−))⊗(⨁n=0∞Kfi(n)),Jz− ≅(⨁n=0∞Kfi(n))⊗(Jz−∩Ti−1(Uz−)). Moreover, we have   Ti−1(Jz−∩Ti(Uz−))=Jz−∩Ti−1(Uz−). □ We set   U¯z=Uz/Jz. (4.8) It is a Hopf algebra by Proposition 4.1. Denote by $$\overline{U}_z^0$$, $$\overline{U}_z^\pm$$, $$\overline{U}_z^{\geqq0}$$, $$\overline{U}_z^{\leqq0}$$, $${}^\sharp \overline{U}_z^{0}$$, $${}^\sharp \overline{U}_z^{\geqq0}$$, $${}^\sharp \overline{U}_z^{\leqq0}$$, $$\overline{U}_{z,\pm\gamma}^\pm$$;($$\gamma\in Q^+$$) the images of $${U}_z^0$$, $${U}_z^\pm$$, $${U}_z^{\geqq0}$$, $${U}_z^{\leqq0}$$, $${}^\sharp U_z^{0}$$, $${}^\sharp {U}_z^{\geqq0}$$, $${}^\sharp {U}_z^{\leqq0}$$, $${U}_{z,\pm\gamma}^\pm$$ under $$U_z\to\overline{U}_z$$, respectively. By the above argument we have   U¯z ≅U¯z+⊗U¯z0⊗U¯z−≅U¯z−⊗U¯z0⊗U¯z+,U¯z≧0 ≅U¯z+⊗U¯z0≅U¯z0⊗U¯z+,U¯z≦0≅U¯z−⊗U¯z0≅U¯z0⊗U¯z−,♯U¯z≧0 ≅U¯z+⊗♯U¯z0≅♯U¯z0⊗U¯z+,♯U¯z≦0≅U¯z−⊗♯U¯z0≅♯U¯z0⊗U¯z−,U¯z0 ≅Uz0=⨁h∈hZKkh,♯U¯z0≅♯Uz0=⨁γ∈QKkγ, and   U¯z±=⨁γ∈Q+U¯z,±γ±,U¯z,±γ±≅Uz,±γ±/Jz,±γ±. (4.9) By definition $$\tau_z$$ induces a bilinear form   τ¯z:♯U¯z≧0×♯U¯z≦0→K such that for any $$\gamma\in Q^+$$ its restriction   τ¯z,γ:U¯z,γ+×U¯z,−γ−→K is non-degenerate. For $$\lambda\in P$$ and a $$\overline{U}_z$$-module $$V$$ we set   Vλ={v∈V∣khv=z⟨λ,h⟩v(h∈hZ)}. We define a category $${\mathcal O}({\overline{U}}_z)$$ as follows. Its objects are $${\overline{U}}_z$$-modules $$V$$ which satisfy   V=⨁λ∈PVλ,dim⁡Vλ<∞(λ∈P), (4.10) and such that there exist finitely many $$\lambda_1,\ldots, \lambda_r\in P$$ such that   {λ∈P∣Vλ≠{0}}⊂⋃k=1r(λk−Q+). The morphisms are homomorphisms of $${\overline{U}}_z$$-modules. We say that a $${\overline{U}}_z$$-module $$V$$ is integrable if $$V=\bigoplus_{\lambda\in P}V_\lambda$$ and for any $$v\in V$$ there exists $$N>0$$ such that for $$i\in I$$ and $$n\geqq N$$ we have $$e_i^{(n)}v=f_i^{(n)}v=0$$. We denote by $${\mathcal O}^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)$$ the full subcategory of $${\mathcal O}({\overline{U}}_z)$$ consisting of integrable $${\overline{U}}_z$$-modules belonging to $${\mathcal O}({\overline{U}}_z)$$. For each coset $$C=\mu+Q\in P/Q$$ we denote by $${\mathcal O}_C({\overline{U}}_z)$$ the full subcategory of $${\mathcal O}({\overline{U}}_z)$$ consisting of $$V\in {\mathcal O}_C({\overline{U}}_z)$$ such that $$V=\bigoplus_{\lambda\in C}V_\lambda$$. We also set $${\mathcal O}_C^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)={\mathcal O}_C({\overline{U}}_z)\cap{\mathcal O}^{\mathop{\rm int}\nolimits}({\overline{U}}_z)$$. Then we have   O(U¯z)=⨁C∈P/QOC(U¯z),Oint(U¯z)=⨁C∈P/QOCint(U¯z). (4.11) For $$\lambda\in P$$ we define $$M_z(\lambda)\in{\mathcal O}_{\lambda+Q}({\overline{U}}_z)$$ by   Mz(λ)=U¯z/(∑h∈hZU¯z(kh−z⟨λ,h⟩)+∑i∈IU¯zei), and for $$\lambda\in P^+$$ we define $$V_z(\lambda)\in{\mathcal O}_{\lambda+Q}^{{\mathop{\rm int}\nolimits}}({\overline{U}}_z)$$ by   Vz(λ)=U¯z/(∑h∈hZU¯z(kh−z⟨λ,h⟩)+∑i∈IU¯zei+∑i∈IU¯zfi(⟨λ,hi⟩+1)). Let $$\lambda\in P$$. A $${\overline{U}}_z$$-module $$V$$ is called a highest weight module with highest weight $$\lambda$$ if there exists $$v\in V_\lambda\setminus\{0\}$$ such that $$V={\overline{U}}_z v$$ and $$xv=\varepsilon(x)v$$;($$x\in {\overline{U}}_z^+$$). Then we have $$V\in{\mathcal O}_{\lambda+Q}({\overline{U}}_z)$$. A $${\overline{U}}_z$$-module is a highest weight module with highest weight $$\lambda$$ if and only if it is a non-zero quotient of $$M_z(\lambda)$$. If there exists an integrable highest weight module with highest weight $$\lambda$$, then we have $$\lambda\in P^+$$. For $$\lambda\in P^+$$ a $${\overline{U}}_z$$-module is an integrable highest weight module with highest weight $$\lambda$$ if and only if it is a non-zero quotient of $$V_z(\lambda)$$. For $$V\in{\mathcal O}({\overline{U}}_z)$$ we define its formal character by   ch⁡(V)=∑λ∈Pdim⁡Vλe(λ)∈E. We have   ch⁡(Mz(λ))=e(λ)D¯−1(λ∈P), where   D¯−1=∑γ∈Q+dim⁡U¯z,−γ−e(−γ)(λ∈P). For each coset $$C=\mu+Q\in P/Q$$ we fix a function $$f_C:C\to {\mathbb{Z}}$$ such that   fC(λ)−fC(λ−αi)=2⟨λ,ti⟩(λ∈C,i∈I). Remark 4.3. The function $$f_C$$ is unique up to addition of a constant function. If we extend $$(\;,\;):E\times E\to{\mathbb{Q}}$$ to a $$W$$-invariant symmetric bilinear form on $${\mathfrak{h}}^*$$, then $$f_C$$ is given by   fC(λ)=(λ+ρ,λ+ρ)+a(λ∈C) for some $$a\in {\mathbb{Q}}$$. □ For $$\gamma\in Q^+$$ let $$ \overline{C}_\gamma\in \overline{U}^+_{z,\gamma}\otimes \overline{U}^-_{z,-\gamma} $$ be the canonical element of the non-degenerate bilinear form $$\overline{\tau}_{z,\gamma}$$. Following Drinfeld we set   Ωγ=(m∘(S⊗1)∘P)(C¯γ)∈U¯z,−γ−U¯z0U¯z,γ+, where $$m:\overline{U}_z\otimes\overline{U}_z\to \overline{U}_z$$ and $$P:\overline{U}_z\otimes \overline{U}_z\to \overline{U}_z\otimes \overline{U}_z$$ are given by $$m(a,b)=ab$$, $$P(a\otimes b)=b\otimes a$$ (see [12, Section 3.2], [11, Section 6.1]). Let $$C\in P/Q$$. For $$V\in{\mathcal O}_C(\overline{U}_z)$$ we define a linear map   Ω:V→V (4.12) by   Ω(v)=zfC(λ)∑γ∈Q+Ωγv(v∈Vλ). This operator is called the quantum Casimir operator. As in [12, Section 3.2] we have the following. Proposition 4.4. Let $$C\in P/Q$$. For $$\lambda\in C$$ the operator $$\Omega$$ acts on $$M_z(\lambda)$$ as $$z^{f_C(\lambda)}\mathop{\rm id}\nolimits$$. □ Since $$z$$ is not a root of 1, we have   zfC(λ)=zfC(μ)⟹fC(λ)=fC(μ). 5 Main Results For $$w\in W$$ and $$x=\sum_{\lambda\in P}c_\lambda e(\lambda)\in{\mathcal E}$$ we set   wx=∑λ∈Pcλe(wλ),w∘x=∑λ∈Pcλe(w∘λ). The elements $$wx$$, $$w\circ x$$ may not belong to $${\mathcal E}$$; however, we will only consider the case where $$wx, w\circ x\in {\mathcal E}$$. We denote by $${\mathop{\rm sgn}\nolimits}:W\to\{\pm1\}$$ the character given by $${\mathop{\rm sgn}\nolimits}(s_i)=-1$$ for $$i\in I$$. Proposition 5.1. For any $$w\in W$$ we have $$w\circ \overline{D}={\mathop{\rm sgn}\nolimits}(w)\overline{D}$$. □ Proof. We may assume that $$w=s_i$$ for $$i\in I$$. Define $$D_i, \overline{D}_i\in{\mathcal E}$$ by   D =(1−e(−αi))Di,D¯=(1−e(−αi))D¯i. Then we have $$D_i=\prod_{\alpha\in\Delta^+\setminus\{\alpha_i\}}(1-e(-\alpha))^{m_\alpha}$$. Moreover, by Lemma 3.2, Lemma 4.2, and (4.9) we have   Di−1 =∑γ∈Q+dim⁡(Uz,−γ−∩Ti(Uz−))e(−γ) =∑γ∈Q+dim⁡(Uz,−γ−∩Ti−1(Uz−))e(−γ),D¯i−1 =Di−1−∑γ∈Q+dim⁡(Jz,−γ−∩Ti(Uz−))e(−γ) =Di−1−∑γ∈Q+dim⁡(Jz,−γ−∩Ti−1(Uz−))e(−γ). By $$s_i\circ\overline{D}=-(1-e(-\alpha_i))s_i\overline{D}_i$$ we have only to show $$s_i\overline{D}_i=\overline{D}_i$$. By Lemma 4.2 we have   si(∑γ∈Q+dim⁡(Jz,−γ−∩Ti(Uz−))e(−γ)) =∑γ∈Q+dim⁡(Jz−∩Ti(Uz,−γ−))e(−γ) =∑γ∈Q+dim⁡(Jz,−γ−∩Ti−1(Uz−))e(−γ), and hence the assertion follows from $$s_iD_i=D_i$$. ■ Proposition 5.2. Let $$\lambda\in P^+$$. Assume that $$V$$ is an integrable highest weight $${\overline{U}}_z$$-module with highest weight $$\lambda$$. Then we have   ch⁡(V)=∑w∈Wsgn(w)ch⁡(Mz(w∘λ)). □ Proof. The proof is the same as the one for Lie algebras in [6, Theorem 10.4]. Set $$C=\lambda+Q\in P/Q$$. Similarly to [6, Proposition 9.8] we have   ch⁡(V)=∑μ∈λ−Q+,fC(μ)=fC(λ)cμch⁡(Mz(μ))(cμ∈Z,cλ=1). (5.1) Multiplying (5.1) by $$\overline{D}$$ we obtain   D¯ch⁡(V)=∑μ∈λ−Q+,fC(μ)=fC(λ)cμe(μ). Using the action of $$T_i\;(i\in I)$$ on $$V$$ we see that $$w\mathop{\rm ch}\nolimits(V)=\mathop{\rm ch}\nolimits(V)$$ for $$w\in W$$, and hence $$w\circ(\overline{D}\mathop{\rm ch}\nolimits(V))={\mathop{\rm sgn}\nolimits}(w)\overline{D}\mathop{\rm ch}\nolimits(V)$$ for any $$w\in W$$. It follows that   cμ=sgn(w)cw∘μ(μ∈λ−Q+,w∈W). (5.2) Assume that $$\mu\in\lambda-Q^+$$ satisfies $$c_\mu\ne0$$. By (5.2) $$W\circ\mu\subset\lambda-Q^+$$, and hence we can take $$\mu'\in W\circ\mu$$ such that $${\mathop{\rm ht}\nolimits}(\lambda-\mu')$$ is minimal, where $${\mathop{\rm ht}\nolimits}(\sum_im_i\alpha_i)=\sum_im_i$$. Then we have $$\langle\mu',h_i\rangle\geqq0$$ for any $$i\in I$$ by $$s_i\circ\mu'= \mu'-(\langle\mu',h_i\rangle+1)\alpha_i$$ and (5.2). Namely, we have $$\mu'\in P^+$$. Then by [6, Lemma 10.3] we obtain $$\mu'=\lambda$$. □ Remark 5.3. Heckenberger pointed out to me that Proposition 5.2 also follows from the existence of the BGG resolution of integrable highest weight modules of quantized enveloping algebras given in [5] □ Recall that any integrable highest weight module $$V$$ with highest weight $$\lambda$$ is a quotient of $$V_z(\lambda)$$. Proposition 5.2 tells us that its character $$\mathop{\rm ch}\nolimits(V)$$ only depends on $$\lambda$$. It follows that any integrable highest weight module with highest weight $$\lambda$$ is isomorphic to $$V_z(\lambda)$$. Consider the case $$\lambda=0$$. Since $$V_z(0)$$ is the trivial one-dimensional module, we obtain the identity   1=(∑w∈Wsgn(w)e(w∘0))(∑γ∈Q+dim⁡U¯z,−γ−e(−γ)) in $${\mathcal E}$$ by Proposition 5.2. On the other hand by the corresponding result for the Kac–Moody Lie algebra we have   1=(∑w∈Wsgn(w)e(w∘0))(∑γ∈Q+dim⁡Uz,−γ−e(−γ)). It follows that $$U^-_{z,-\gamma}\cong{\overline{U}}^-_{z,-\gamma}$$ for any $$\gamma \in Q^+$$. By $$\dim U^-_{z,-\gamma}=\dim U^+_{z,\gamma}$$ and the non-degeneracy of $$\overline{\tau}_{z,\gamma}$$ we also have $$U^+_{z,\gamma}\cong{\overline{U}}^+_{z,\gamma}$$ for any $$\gamma\in Q^+$$. We have obtained the following results. Theorem 5.4 The Drinfeld pairing   τz,γ:Uz,γ+×Uz,−γ−→K is non-degenerate for any $$\gamma\in Q^+$$. □ Theorem 5.5 Let $$\lambda\in P^+$$. Assume that $$V$$ is an integrable highest weight $$U_z$$-module with highest weight $$\lambda$$. Then we have   ch⁡(V)=D−1∑w∈Wsgn(w)e(w∘λ). □ By Theorem 5.4 we can define the quantum Casimir operator $$\Omega$$ for $$U_z$$. As in [11, Section 6.2] we have the following. Theorem 5.6 Any object of $${\mathcal O}^{{\mathop{\rm int}\nolimits}}(U_z)$$ is a direct sum of $$V_z(\lambda)$$’s for $$\lambda\in P^+$$. □ By Theorem 5.4 we have the following. Theorem 5.7 Let $$\gamma\in Q^+$$. Take bases $$\{x_r\}$$ and $$\{y_s\}$$ of $$U^+_{{\mathbb{A}},\gamma}$$ and $$U^-_{{\mathbb{A}},-\gamma}$$, respectively, and set $$f_\gamma=\det(\tau_{{\tilde{{\mathbb{A}}}},\gamma}(x_r,y_s))_{r,s}$$. Then we have $$f_\gamma\in{\tilde{{\mathbb{A}}}}^\times$$. Namely, we have   fγ=±qaf1±1⋯fN±1, where $$a\in{\mathbb{Z}}$$, and $$f_1,\dots, f_N\in{\mathbb{Z}}[q]$$ are cyclotomic polynomials. □ Proof. We can write $$f_\gamma=mgh$$, where $$m\in{\mathbb{Z}}_{>0}$$, $$g\in {\mathbb{Z}}[q]$$ is a primitive polynomial with $$g(0)>0$$ whose irreducible factor is not cyclotomic, and $$h\in{\tilde{{\mathbb{A}}}}^\times$$. For any field $$K$$ and $$z\in K^\times$$ which is not a root of 1, the specialization of $$f_\gamma$$ with respect to the ring homomorphism $${\tilde{{\mathbb{A}}}}\to K\;(q\mapsto z)$$ is non-zero by Theorem 5.4. Hence we see easily that $$m=1$$ and $$g=1$$. ■ In the finite case Theorem 5.7 is well-known [8, 9, 11]. In the affine case this is a consequence of [3, 4], where $$\det(\tau_{{\tilde{{\mathbb{A}}}},\gamma}(x_r,y_s))_{r,s}$$ is determined explicitly by a case-by-case calculation. Funding This work was supported by Grants-in-Aid for Scientific Research (C) 15K04790 from Japan Society for the Promotion of Science. Acknowledgement The first draft of this paper contained only results when $$K$$ is of characteristic zero. 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Google Scholar CrossRef Search ADS   [4] Damiani I. “The R-matrix for (twisted) affine quantum algebras.” In Representations and Quantizations (Shanghai, 1998) , edited by Wang J. and Lin Z. 89– 144, Beijing: Higher Education Press, 2000. [5] Heckenberger I. and Kolb. S. “On the Bernstein-Gelfand-Gelfand resolution for Kac–Moody algebras and quantized enveloping algebras.” Transformation Groups  12, no. 4 ( 2007): 647– 55. Google Scholar CrossRef Search ADS   [6] Kac V. Infinite Dimensional Lie Algebras , 3rd ed. Cambridge: Cambridge University Press, 1990. Google Scholar CrossRef Search ADS   [7] Kashiwara M. “On crystal bases.” In Representations of Groups (Banff, AB, 1994) , edited by Allison B. and Cliff G. 155– 97. CMS Conference Proceedings 16. Providence, RI: American Mathematical Society, 1995. [8] Kirillov A. N. and Reshetikhin. N. “q-Weyl group and a multiplicative formula for universal R-matrices.” Communications in Mathematical Physics  134, no. 2 ( 1990): 421– 31. Google Scholar CrossRef Search ADS   [9] Levendorskii S. Z. and Soibelman. Ya. S. “Some applications of the quantum Weyl groups.” Journal of Geometry and Physics  7, no. 2 ( 1990): 241– 54. Google Scholar CrossRef Search ADS   [10] Lusztig G. “Quantum deformations of certain simple modules over enveloping algebras.” Advances in Mathematics  70, no. 2 ( 1988): 237– 49. Google Scholar CrossRef Search ADS   [11] Lusztig G. Introduction to Quantum Groups . Progress in Mathematics 110. Boston: Birkhäuser, 1993. [12] Tanisaki T. “Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras.” International Journal of Modern Physics A7, Suppl . 1B ( 1992): 941– 61. [13] Tanisaki T. “Modules over quantized coordinate algebras and PBW-bases.” Journal of the Mathematical Society of Japan . arXiv:1409.7973 (in press). [14] Tanisaki T. “Invariance of the Drinfeld pairing of a quantum group.” Tokyo Journal of Mathematics . arXiv:1503.04573 (in press). [15] Tsuchioka S. “Graded Cartan determinants of the symmetric groups.” Transactions of the American Mathematical Society  366, no. 4 ( 2014): 2019– 40. Google Scholar CrossRef Search ADS   Communicated by Prof. Masaki Kashiwara © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

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Published: Feb 1, 2018

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