Crystals and affine Hecke algebras of type D. (English) Zbl 1206.17014

Summary: The Lascoux-Leclerc-Thibon-Ariki theory asserts that the \(K\)-group of the representations of the affine Hecke algebras of type \(A\) is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra \(\mathfrak{g}\) where \(\mathfrak{g}\) is \(\mathfrak{gl}_{\infty}\) or the affine Lie algebra \(A^{(1)}_{\ell}\), and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author [Proc. Japan Acad., Ser. A 82, No. 8, 131–136 (2006; Zbl 1130.20008)] presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type \(B\). In this note, we present similar conjectures for certain classes of irreducible representations of affine Hecke algebras of type \(D\). The crystal for type \(D\) is a double cover of the one for type \(B\).


17B37 Quantum groups (quantized enveloping algebras) and related deformations
20C08 Hecke algebras and their representations
20G05 Representation theory for linear algebraic groups


Zbl 1130.20008
Full Text: DOI arXiv Euclid


\(V_{\theta}(\lambda)\) has a crystal basis and an upper global basis.
\(V_{\theta}\) has a crystal basis and an upper global basis.
\(\mathrm{K}^{\text{D}}\) is isomorphic to a specialization of \(V_{\theta}\) at \(q=1\), and the irreducible representations correspond to the upper global basis of \(V_{\theta}\) at \(q=1\). \endenumerate The representations of \(\mathrm{H}^{\text{D}}_{n}\) such that some of \(X_{i}\) have an eigenvalue \(\pm1\) are again excluded. Note that the crystal basis for type D is a double cover of the one for type B. \sectionSymmetric crystals. \subsectionQuantized universal enveloping algebras. We shall recall the quantized universal enveloping algebra \(U_{q}(\mathfrak{g})\). Let \(I\) be an index set (for simple roots), and \(Q\) the free \(\mathbf{Z}\)-module with a basis \(\{\alpha_{i}\}_{i\in I}\). Let \(({\bullet},{\bullet}): Q\times Q\to\mathbf{Z}\) be a symmetric bilinear form such that \((\alpha_{i},\alpha_{i})/2\in\mathbf{Z}_{>0}\) for any \(i\) and \((\alpha_{i}^{\vee},\alpha_{j})\in\mathbf{Z}_{\leqslant0}\) for \(i\neq j\) where \(\alpha_{i}^{\vee}:=2\alpha_{i}/(\alpha_{i},\alpha_{i})\). Let \(q\) be an indeterminate and set \(\mathbf{K}:=\mathbf{Q}(q)\). We define its subrings \(\mathbf{A}_{0}\), \(\mathbf{A}_{\infty}\) and \(\mathbf{A}\) as follows: \beginalign* \mathbfA_0 & =\f/g;f(q),g(q)\in \mathbfQ[q], g(0)\neq 0, \mathbfA_\infty & =\f/g;f(q^-1),g(q^-1)\in \mathbfQ[q^-1], g(0)\neq 0, \mathbfA & =\mathbfQ[q,q^-1]. \endalign* \begindefn \labelU__q__g_ The quantized universal enveloping algebra \(U_{q}(\mathfrak{g})\) is the \(\mathbf{K}\)-algebra generated by the elements \(e_{i},f_{i}\) and invertible elements \(t_{i}\;(i\in I)\) with the following defining relations. \beginenumerate
The \(t_{i}\)’s commute with each other.
\(t_{j}e_{i}\,t_{j}^{-1}=q^{(\alpha_{j},\alpha_{i})}\,e_{i}\) and \(t_{j}f_{i}t_{j}^{-1}=q^{-(\alpha_{j},\alpha_{i})}f_{i}\) for any \(i,j\in I\).
\([ e_{i},f_{j}]=\delta_{ij}\dfrac{t_{i}-t_{i}^{-1}}{q_{i}-q_{i}^{-1}}\) for \(i\), \(j\in I\), where \(q_{i}:= q^{(\alpha_{i},\alpha_{i})/2}\).
(Serre relation) For \(i\neq j\), \beginequation* \sum^b_k=0(-1)^ke^(k)_ie_je^(b-k)_i=\sum^b_k=0(-1)^kf^(k)_i f_jf_i^(b-k)=0. \endequation* Here \(b=1-(\alpha_{i}^{\vee},\alpha_{j})\) and \beginalign* e^(k)_i & =e^k_i/[ k]_i!, f^(k)_i=f^k_i/[ k]_i!, [ k]_i & =(q^k_i-q^-k_i)/(q_i-q^-1_i), [ k]_i!=[ 1]_i\cdots [ k]_i. \endalign* \endenumerate \enddefn Let us denote by \( U^{-}_{q}(\mathfrak{g})\) (resp. \(U^{+}_{q}(\mathfrak{g})\)) the subalgebra of \(U_{q}(\mathfrak{g})\) generated by the \(f_{i}\)’s (resp. the \(e_{i}\)’s). Let \(e'_{i}\) and \(e^*_{i}\) be the operators on \( U^{-}_{q}(\mathfrak{g})\) defined by \beginequation* [e_i,a]=\frac(e^*_ia)t_i-t_i^-1e’_iaq_i-q_i^-1(a\in U^-_q(\mathfrakg)). \endequation* Then these operators satisfy the following formula similar to derivations: \beginalign* e_i’(ab) & =e_i’(a)b+(\mathrmAd(t_i)a)e_i’b, e_i^*(ab) & =ae_i^*b+(e_i^*a)(\mathrmAd(t_i)b). \endalign* The algebra \( U^{-}_{q}(\mathfrak{g})\) has a unique symmetric bilinear form \(({\bullet},{\bullet})\) such that \((1,1)=1\) and \beginequation* (e’_ia,b)=(a,f_ib)for any \(a,b\in U^{-}_{q}(\mathfrak{g})\). \endequation* It is non-degenerate and satisfies \((e^*_{i}a,b)=(a,bf_{i})\). \subsectionSymmetry. \labelsubsec_symmetry Let \(\theta\) be an automorphism of \(I\) such that \(\theta^{2}=\mathop{\mathrm{id}}\) and \((\alpha_{\theta(i)},\alpha_{\theta(j)})=(\alpha_{i},\alpha_{j})\). Hence it extends to an automorphism of the root lattice \(Q\) by \(\theta(\alpha_{i})=\alpha_{\theta(i)}\), and induces an automorphism of \(U_{q}(\mathfrak{g})\). Let \(\mathcal{B}_{\theta}(\mathfrak{g})\) be the \(\mathbf{K}\)-algebra generated by \(E_{i}\), \(F_{i}\), and invertible elements \(K_{i}\) (\(i\in I\)) satisfying the following defining relations: \beginequation \Biggl beginarrayrl (i) & the \(K_{i}\)’s commute with each other, (ii) & \(K_{\theta(i)}=K_{i}\) for any \(i\in I\), (iii) & \(K_{i}E_{j}K_{i}^{-1}=q^{(\alpha_{i}+\alpha_{\theta(i)},\alpha_{j})}E_{j}\) & and \(K_{i}F_{j}K_{i}^{-1}=q^{(\alpha_{i}+\alpha_{\theta(i)},-\alpha_{j})}F_{j}\) & for \(i,j\in I\), (iv) & \(E_{i}F_{j}=q^{-(\alpha_{i},\alpha_{j})}F_{j}E_{i}+(\delta_{i,j}+\delta_{\theta(i),j}K_{i})\) & for \(i,j\in I\), (v) & the \(E_{i}\)’s and the \(F_{i}\)’s satisfy the Serre & relations. \endarray \labelrel_EFK \endequation Hence \(\mathcal{B}_{\theta}(\mathfrak{g})\simeq U^{-}_{q}(\mathfrak{g})\otimes\mathbf{K}[K_{i}^{\pm1};i\in I]\otimes U^{+}_{q}(\mathfrak{g})\). We set \(E_{i}^{(n)}=E_{i}^{n}/[n]_{i}!\) and \(F_{i}^{(n)}=F_{i}^{n}/[n]_{i}!\). \beginprop \labelprop_Vtheta
There exists a \(\mathcal{B}_{\theta}(\mathfrak{g})\)-module \(V_{\theta}\) generated by linearly independent vectors \(\phi_{+}\) and \(\phi_{-}\) such that \beginenumerate
\(E_{i}\phi_{\pm}=0\) for any \(i\in I\),
\(K_{i}\phi_{\pm}=\phi_{\mp}\) for any \(i\in I\),
\(\{u\in V_{\theta};E_{i}u=0\;\textit{for any}\;i\in I\}\) \(= \mathbf{K}\phi_{+} \oplus \mathbf{K}\phi_{-}\). \endenumerate
Moreover such a \(V_{\theta}\) is unique up to an isomorphism.
There exists a unique symmetric bilinear form \(({\bullet},{\bullet})\) on \(V_{\theta}\) such that \((\phi_{\varepsilon_{1}},\phi_{\varepsilon_{2}})=\delta_{\varepsilon_{1},\varepsilon_{2}}\) for \(\varepsilon_{1},\varepsilon_{2} \in \{+,-\}\) and \((E_{i}u,v)=(u,F_{i}v)\) for any \(i\in I\) and \(u,v\in V_{\theta}\), and it is non-degenerate. \endenumerate \endprop Such a \(V_{\theta}\) is constructed as follows. Let \(\mathcal{S}\) be the quantum shuffle algebra (see [Leclerc?]) generated by words \(\langle i_{1}, \dots,i_{l} \rangle\) for \(i_{1}, \dots,i_{l} \in I\) and \(l \geq 1\) and \(\phi_{+}''\) and \(\phi_{-}''\) as two empty words. We assign to a word \(\langle i_{1}, \dots, i_{l} \rangle \) the weight \(-(\alpha_{i_{1}}+ \cdots +\alpha_{i_{l}})\). We define the actions of \(E_{i}\), \(F_{i}\) and \(K_{i}\) on \(\mathcal{S}\) as follows: \beginalign* & F_i \phi”_+= \langle i \rangle, F_i \phi”_-= \langle \theta i \rangle, & E_i \langle j \rangle = \delta_i,j\phi”_+ +\delta_i,\theta j\phi_-”, & K_i \phi”_\pm = \phi”_\mp, & K_i\langle i_1, …, i_l \rangle=q^-(\alpha_i+\alpha_\theta(i),\alpha_i_1+\cdots+\alpha_i_l) &     \cdot\langle i_1,…,i_l-1, \theta(i_l) \rangle, & E_i\langle i_1, …, i_l \rangle= \delta_i,i_1\langle i_2,…, i_l \rangle, & F_i \langle i_1,...,i_l\rangle= \langle i \rangle *\langle i_1,...,i_l\rangle &     +q^(\alpha_i,\mathrmwt(\langle i_1,...,i_l-1,\theta(i_l) \rangle))\langle i_1,...,i_l-1,\theta(i_l)\rangle * \langle \theta i \rangle & =\sum_\nu=0^lq^-(\alpha_i,\alpha_i_1+\cdots+\alpha_i_\nu)\langle i_1,...,i_\nu,i,i_\nu+1,...,i_l\rangle &     + q^-(\alpha_i,\alpha_1+\cdots +\alpha_i_l-1+\alpha_\theta(i_l)) & \sum_\nu=0^l q^-(\alpha_\theta(i),\alpha_\nu+1+\cdots+\alpha_i_l-1+\alpha_\theta(i_l)) &     \cdot\langle i_1,...,i_\nu,\theta(i),i_\nu+1,...,i_l-1,\theta(i_l)\rangle \endalign* for \(i,j \in I\), \(l\geqslant1\) and \(i_{1},\ldots,i_{l}\in I\). Then the operators \(E_{i}\), \(F_{i}\) and \(K_{i}\) satisfy the commutation relations \eqrefrel_EFK except the Serre relations for the \(E_{i}\)’s. Consider the \(U_{q}^{-}(\mathfrak{g})\)-module \(V'=U_{q}^{-}(\mathfrak{g}) \phi'_{+} \oplus U_{q}^{-}(\mathfrak{g}) \phi'_{-}\) generated by a pair of vacuum vectors \(\phi'_{\pm}\). There exists a unique \(U_{q}^{-}(\mathfrak{g})\)-linear map \(\psi: V' \rightarrow \mathcal{S}\) such that \(\phi'_{\pm} \mapsto \phi''_{\pm}\). We define an action of \(\mathcal{B}_{\theta}(\mathfrak{g})\) on \(V'\) by \beginalign* K_i (a \phi’_\pm) & = (\mathrmAd(t_it_\theta(i))a)\phi’_\mp, E_i (a \phi’_\pm) & = e_i’(a)\phi’_\pm + Ad(t_i)(e_\theta i^*(a)) \phi’_\mp, F_i (a \phi’_\pm) & = f_ia \phi’_\pm \endalign* for \(a\in U^{-}_{q}(\mathfrak{g})\). Then \(\psi\) commutes with the actions of \(E_{i}\), \(F_{i}\) and \(K_{i}\), and its image \(\psi(V')\) is \(V_{\theta}\). Hereafter we assume further that \beginequation there is no \(i\in I\) such that \(\theta(i)=i\). \labele2_2 \endequation Under this condition, we conjecture that \(V_{\theta}\) has a crystal basis. This means the following. We define the modified root operators: \beginequation* \widetildeE_i(u)=\sum_n\geqslant1F_i^(n-1)u_n and \widetildeF_i(u)=\sum_n\geqslant0F_i^(n+1)u_n \endequation* when writing \(u=\sum_{n\geqslant0}F_{i}^{(n)}u_{n}\) with \(E_{i}u_{n}=0\). Let \(L_{\theta}\) be the \(\mathbf{A}_{0}\)-submodule of \(V_{\theta}\) generated by \(\widetilde{F}_{i_{1}}\cdots\widetilde{F}_{i_{\ell}}\phi_{\pm}\) (\(\ell\geqslant0\) and \(i_{1},\ldots,i_{\ell}\in I\),), and define the subset \( B_{\theta}\subset L_{\theta}/q L_{\theta}\) by: \beginequation* B_\theta:= widetildeF_i_1\cdots\widetildeF_i_\ell\phi_\pm\bmod q L_\theta;\ell\geqslant0, i_1,..., i_\ell\in I. \endequation* \beginconj \labelconj_crystal \beginenumerate
\(\widetilde{F}_{i}L_{\theta}\subset L_{\theta}\) and \(\widetilde{E}_{i}L_{\theta}\subset L_{\theta}\),
\(B_{\theta}\) is a basis of \(L_{\theta}/qL_{\theta}\),
\(\widetilde{F}_{i} B_{\theta}\subset B_{\theta}\), and \(\widetilde{E}_{i} B_{\theta}\subset B_{\theta}\sqcup\{0\}\). \endenumerate \endconj Moreover we conjecture that \(V_{\theta}\) has a global crystal basis. Namely, let \(-\) be the bar-operator of \(V_{\theta}\), which is characterized by: \(\overline{q}=q^{-1}\), \(-\) commutes with the \(E_{i}\)’s, and \((\phi_{\pm})^{-}=\phi_{\pm}\) (such an operator exists). Let us denote by \(\mathcal{B}_{\theta}(\mathfrak{g})_{\mathbf{A}}^{\text{up}}\) the \(\mathbf{A}\)-subalgebra of \(\mathcal{B}_{\theta}(\mathfrak{g})\) generated by \(E_{i}^{(n)}\), \(F_{i}\) and \(K_{i}^{\pm1}\) (\(i \in I\)). Let \((V_{\theta})_{\mathbf{A}}\) be the largest \(\mathcal{B}_{\theta}(\mathfrak{g})^{\text{up}}_{\mathbf{A}}\)-submodule of \(V_{\theta}\) such that \((V_{\theta})_{\mathbf{A}}\cap(\mathbf{K}\phi_{+}+\mathbf{K}\phi_{-})=\mathbf{A}\phi_{+}+\mathbf{A}\phi_{-}\). \beginconj \labelconj_balanced \((L_{\theta},L_{\theta}^{-},(V_{\theta})_{\mathbf{A}})\) is balanced. \endconj Namely, \(E:= L_{\theta}\cap L_{\theta}^{-}\cap(V_{\theta})_{\mathbf{A}}\to L_{\theta}/qL_{\theta}\) is an isomorphism. Let \(G^{\text{up}}\colon L_{\theta}/q L_{\theta}\stackrel{\sim}{\longrightarrow} E\) be its inverse. Then \(\{G^{\text{up}}(b);b\in B_{\theta}\}\) forms a basis of \(V_{\theta}\). We call this basis the upper global basis of \(V_{\theta}\). \beginrmk \labelrem_crystal Assume that Conjectures 2.3 and 2.3 hold. \beginenumerate
We have \(\{b\in B_{\theta};\widetilde{E}_{i}b=0\;\text{for any}\;i\in I\} =\{\phi_{+},\phi_{-}\}\).
There exists a unique involution \(\sigma\) of the \(\mathcal{B}_{\theta}(\mathfrak{g})\)-module \(V_{\theta}\) such that \(\sigma(\phi_{\pm})=\phi_{\mp}\). It extends to the involution \(\sigma\) of \(\mathcal{S}\) by \(\sigma(\langle i_{1},\ldots,i_{l}\rangle)=\langle i_{1},\ldots,i_{l-1},\theta(i_{l})\rangle\). It induces also involutions of \(L_{\theta}\) and \( B_{\theta}\).
We have \(\sigma(b)\neq b\) for any \(b\in B_{\theta}\).
We conjecture that \(\widetilde{F}_{i}b\neq \widetilde{F}_{j}b\) for any \(b\in B_{\theta}\) and \(i\neq j\in I\).
In [EK?], a \(\mathcal{B}_{\theta}(\mathfrak{g})\)-module \(V_{\theta}(\lambda)=\mathcal{B}_{\theta}(\mathfrak{g})\phi_{\lambda}\) and its crystal basis \(B_{\theta}(\lambda)\) are introduced. We have a monomorphism of \(\mathcal{B}_{\theta}(\mathfrak{g})\)-modules \beginequation* \iota: V_\theta(\lambda)\rightarrowtail V_\theta \endequation* with \(\lambda=0\), which sends \(\phi_{\lambda}\) to \(\phi_{+}+\phi_{-}\). Its image coincides with \(\{v\in V_{\theta};\sigma(v)=v\}\). Any element \(b\in B_{\theta}(\lambda)\) is sent to \(b'+\sigma(b')\) for some \(b'\in B_{\theta}\). Moreover, we have \(\iota(G^{\text{up}}(b))=G^{\text{up}}(b')+\sigma (G^{\text{up}}(b'))\). In particular, we have \beginequation* B_\theta(\lambda)\simeq B_\theta/\sim. \endequation* Here \(\sim\) is the equivalence relation given by \(b\sim\sigma b\). \endenumerate \endrmk \sectionAffine Hecke algebra of type D. \subsectionDefinition. \labelsubsec_def For \(p \in\mathbf{C}^{*}\) and \(n\in\mathbf{Z}_{\geqslant2}\), the affine Hecke algebra \(\mathrm{H}^{\text{D}}_{n}\) of type \(D_{n}\) is the \(\mathbf{C}\)-algebra generated by \(T_{i}\) (\(0\leqslant i<n\)) and invertible elements \(X_{i}\) (\(1\leqslant i\leqslant n\)) satisfying the defining relations: \beginenumerate
the \(X_{i}\)’s commute with each other,
the \(T_{i}\)’s satisfy the braid relation: \(T_{1}T_{0}=T_{0}T_{1}\), \(T_{0}T_{2}T_{0}=T_{2}T_{0}T_{2}\), \(T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}\) (\(1\leqslant i<n-1\)), \(T_{i}T_{j}=T_{j}T_{i}\) (\(1\leqslant i<j-1<n-1\) or \(i=0<3\leqslant j<n\)),
· Zbl 1135.54011
\((T_{i}-p)(T_{i}+p^{-1})=0\) (\(0\leqslant i<n\)),
\(T_{0}X_{1}^{-1}T_{0}=X_{2}\), \(T_{i}X_{i}T_{i}=X_{i+1}\) (\(1\leqslant i<n\)), and \(T_{i}X_{j}=X_{j}T_{i}\) if \(1 \leq i\neq j,j-1\) or \(i=0\) and \(j \geq 3\). \endenumerate We define \(\mathrm{H}^{\text{D}}_{0}=\mathbf{C} \oplus \mathbf{C}\) and \(\mathrm{H}^{\text{D}}_{1}=\mathbf{C}[X_{1}^{\pm1}]\). We assume that \(p\in\mathbf{C}^{*}\) satisfies \beginequation p^2\neq 1. \labele3_1 \endequation Let us denote by \(\mathbf{P}\mathrm{ol}_{n}\) the Laurent polynomial ring \(\mathbf{C}[X_{1}^{\pm1},\ldots,X_{n}^{\pm1}]\), and by \(\widetilde{\mathbf{P}\mathrm{ol}}_{n}\) its quotient field \(\mathbf{C}(X_{1},\ldots, X_{n})\). Then \(\mathrm{H}^{\text{D}}_{n}\) is isomorphic to the tensor product of \(\mathbf{P}\mathrm{ol}_{n}\) and the subalgebra generated by the \(T_{i}\)’s that is isomorphic to the Hecke algebra of type \(D_{n}\). We have \beginequation* T_ia=(s_ia)T_i+(p-p^-1)\dfraca-s_ia1-X^-\alpha_i^\vee for a\in\mathbfP\mathrmol_n. \endequation* Here, \(X^{-\alpha_{i}^{\vee}}=X_{1}^{-1}X_{2}^{-1}\) (\(i=0\)) and \(X^{-\alpha_{i}^{\vee}}=X_{i}X_{i+1}^{-1}\) (\(1\leqslant i<n\)). The \(s_{i}\)’s are the Weyl group action on \(\mathbf{P}\mathrm{ol}_{n}\): \((s_{0}a)(X_{1},\ldots,X_{n})=a(X_{2}^{-1},X_{1}^{-1},\ldots,X_{n})\) and \((s_{i}a)(X_{1},\ldots,X_{n})=a(X_{1},\ldots,X_{i+1},X_{i},\ldots,X_{n})\) for \(1\leqslant i<n\). \subsectionIntertwiner. The algebra \(\mathrm{H}^{\text{D}}_{n}\) acts faithfully on \(\mathrm{H}^{\text{D}}_{n}/\sum_{i}\mathrm{H}^{\text{D}}_{n}(T_{i}-p)\simeq\mathbf{P}\mathrm{ol}_{n}\). Set \(\varphi_{i}=(1-X^{-\alpha_{i}^{\vee}})T_{i}-(p-p^{-1})\in\mathrm{H}^{\text{D}}_{n}\) and \(\tilde{\varphi}_{i}=(p^{-1}-pX^{-\alpha_{i}^{\vee}})^{-1}\varphi_{i}\in\widetilde{\mathbf{P}\mathrm{ol}}_{n}\otimes_{\mathbf{P}\mathrm{ol}_{n}}\mathrm{H}^{\text{D}}_{n}\). Then the action of \(\tilde{\varphi}_{i}\) on \(\mathbf{P}\mathrm{ol}_{n}\) coincides with \(s_{i}\). They are called intertwiners. \subsectionAffine Hecke algebra of type A. \labelsubsec_affA The affine Hecke algebra \(\mathrm{H}^{\text{A}}_{n}\) of type \(A_{n}\) is isomorphic to the subalgebra of \(\mathrm{H}^{\text{D}}_{n}\) generated by \(T_{i}\) (\(1\leqslant i<n\)) and \(X_{i}^{\pm1}\) (\(1\leqslant i\leqslant n\)). For a finite-dimensional \(\mathrm{H}^{\text{A}}_{n}\)-module \(M\), let us decompose \beginequation M=\textstyle\bigoplus\limits_a\in(\mathbfC^*)^nM_a \labeleq_wtdec \endequation where \beginequation* \beginsplit M_a & =ĭn M;(X_i-a_i)^Nu=0 & for any i and N\gg0 \endsplit \endequation* for \(a=(a_{1},\ldots,a_{n})\in(\mathbf{C}^*)^{n}\). For a subset \(I\subset\mathbf{C}^{*}\), we say that \(M\) is of type \(I\) if all the eigenvalues of \(X_{i}\) belong to \(I\). The group \(\mathbf{Z}\) acts on \(\mathbf{C}^{*}\) by \(\mathbf{Z}\ni n: a\mapsto ap^{2n}\). By well-known results in type A, it is enough to treat the irreducible modules of type \(I\) for an orbit \(I\) with respect to the \(\mathbf{Z}\)-action on \(\mathbf{C}^{*}\) in order to study the irreducible modules over the affine Hecke algebras of type A. \subsectionRepresentations of affine Hecke lgebras of type D. \labelsubsec_main For \(n,m\geqslant 0\), set \(\mathbf{F}_{n,m}:=\mathbf{C}[X_{1}^{\pm1},\ldots,X_{n+m}^{\pm1},D^{-1}]\) where \beginalign* D & :=\prod\limits_1\leqslant i\leqslant n<j\leqslant n+m(X_i-p^2 X_j)(X_i-p^-2X_j) & \cdot(X_i-p^2 X_j^-1)(X_i-p^-2X_j^-1) & \cdot(X_i-X_j)(X_i-X_j^-1). \endalign* Then we can embed \(\mathrm{H}^{\text{D}}_{m}\) into \(\mathrm{H}^{\text{D}}_{n+m}\otimes_{\mathbf{P}\mathrm{ol}_{n+m}}\mathbf{F}_{n,m}\) by \beginalign* T_0 & \mapsto \tilde\varphi_n\cdots\tilde\varphi_1\tilde\varphi_n+1\cdots \tilde\varphi_2T_0\tilde\varphi_2\cdots\tilde\varphi_n+1\tilde\varphi_1\cdots\tilde\varphi_n, T_i & \mapsto T_i+n(1\leqslant i<m), X_i & \mapsto X_i+n(1\leqslant i\leqslant m). \endalign* Its image commutes with \(\mathrm{H}^{\text{D}}_{n}\subset\mathrm{H}^{\text{D}}_{n+m}\). Hence \(\mathrm{H}^{\text{D}}_{n+m}\otimes_{\mathbf{P}\mathrm{ol}_{n+m}\mathbf{F}_{n,m}} is a right \)\mathrm{H}^{D}_{n}\otimes\mathrm{H}^{D}_{m}\(-module. For a finite-dimensional \)\mathrm{H}^{D}_{n}\(-module \)M\(, we decompose \)M\( as in \eqrefeq_wtdec. The semidirect product group \)\mathbf{Z}_{2}\times\mathbf{Z}={1,-1}\times\mathbf{Z}\( acts on \)\mathbf{C}^{*}\( by \)(\epsilon,n): a\mapsto a^{\epsilon} p^{2n}\(. Let \)I\( and \)J\( be \)\mathbf{Z}_{2}\times\mathbf{Z}\(-invariant subsets of \)\mathbf{C}^{*}\( such that \)I\cap J=\emptyset\(. Then for an \)\mathrm{H}^{D}_{n}\(-module \)N\( of type \)I\( and \)\mathrm{H}^{D}_{m}\(-module \)M\( of type \)J\(, the action of \)\mathbf{P}\mathrm{ol}_{n+m}\( on \)N\otimes M\( extends to an action of \)\mathbf{F}_{n,m}\(. We set \beginequation* \beginsplit & N\diamond M & := (\mathrmH^D_n+m\otimes_\mathbfP\mathrmol_n+m\mathbfF_n,m) \underset(\mathrmH^D_n\otimes\mathrmH^D_m)\otimes_\mathbfP\mathrmol_n+m\mathbfF_n,m\otimes (N\otimes M). \endsplit \endequation* \beginlem \beginenumerate\)
Let \(N\) be an irreducible \(\mathrm{H}^{\text{D}}_{n}\)-module of type \(I\) and \(M\) an irreducible \(\mathrm{H}^{\text{D}}_{m}\)-module of type \(J\). Then \(N\diamond M\) is an irreducible \(\mathrm{H}^{\text{D}}_{n+m}\)-module of type \(I\cup J\).
Conversely if \(L\) is an irreducible \(\mathrm{H}^{\text{D}}_{n}\)-module of type \(I\cup J\), then there exists an integer \(m\) \((0\leqslant m\leqslant n)\), an irreducible \(\mathrm{H}^{\text{D}}_{m}\)-module \(N\) of type \(I\) and an irreducible \(\mathrm{H}^{\text{D}}_{n-m}\)-module \(M\) of type \(J\) such that \(L\simeq N\diamond M\).
Assume that a \(\mathbf{Z}_{2}\times\mathbf{Z}\)-orbit \(I\) decomposes into \(I=I_{+}\sqcup I_{-}\) where \(I_{\pm}\) are \(\mathbf{Z}\)-orbits and \(I_{-}=(I_{+})^{-1}\). Then for any irreducible \(\mathrm{H}^{\text{D}}_{n}\)-module \(L\) of type \(I\), there exists an irreducible \(\mathrm{H}^{\text{A}}_{n}\)-module \(M\) such that \(L\simeq\mathop{\mathrm{Ind}}_{\mathrm{H}^{\text{A}}_{n}}^{\mathrm{H}^{\text{D}}_{n}}M\). \endenumerate \endlem Hence in order to study \(\mathrm{H}^{\text{D}}\)-modules, it is enough to study irreducible modules of type \(I\) for a \(\mathbf{Z}_{2}\times\mathbf{Z}\)-orbit \(I\) in \(\mathbf{C}^{*}\) such that \(I\) is a \(\mathbf{Z}\)-orbit, namely \(I=\pm\{p^{n};n\in\mathbf{Z}_{\text{odd}}\}\) or \(I=\pm\{p^{n};n\in\mathbf{Z}_{\text{even}}\}\). For a \(\mathbf{Z}_{2}\times\mathbf{Z}\)-invariant subset \(I\) of \(\mathbf{C}^{*}\), we define \(\mathrm{K}^{\text{D}}_{I,n}\) to be the Grothendieck group of the abelian category of finite-dimensional \(\mathrm{H}^{\text{D}}_{n}\)-modules of type \(I\). We set \(\mathrm{K}^{\text{D}}_{I}=\bigoplus_{n\geqslant0}\mathrm{K}^{\text{D}}_{I,n}\). We take the case \beginequation* I=\p^n;n\in\mathbfZ_odd \endequation* and assume that any of \(\pm1\) is not contained in \(I\). The set \(I\) may be regarded as the set of vertices of a Dynkin diagram. Let us define an automorphism \(\theta\) of \(I\) by \(a\mapsto a^{-1}\). Let \(\mathfrak{g}_{I}\) be the associated Lie algebra (\(\mathfrak{g}_{I}\) is isomorphic to \(\mathfrak{gl}_{\infty}\) if \(p\) has an infinite order, and isomorphic to \(A^{(1)}_{\ell}\) if \(p^{2}\) is a primitive \(\ell\)-th root of unity). For a finite-dimensional \(\mathrm{H}^{\text{D}}_{n}\)-module \(M\) and \(a\in I\), let \(E_{a}M\) be the generalized \(a\)-eigenspace of \(X_{n}\) on \(M\), regarded as an \(\mathrm{H}^{\text{D}}_{n-1}\)-module. Let \(F_{a}M\) be the \(\mathrm{H}^{\text{D}}_{n+1}\)-module \(\mathop{\mathrm{Ind}}_{\mathrm{H}^{\text{D}}_{n}\otimes\mathbf{C}[X_{n+1}^{\pm}]}^{\mathrm{H}^{\text{D}}_{n+1}}(M \otimes (a))\) where \((a)\) is the \(1\)-dimensional representation of \(\mathbf{C}[X_{n+1}^{\pm 1}]\) on which \(X_{n+1}\) acts as \(a\). Then \(E_{a}\) and \(F_{a}\) are exact functors and define \(E_{a}: \mathrm{K}^{\text{D}}_{I,n}\to\mathrm{K}^{\text{D}}_{I,n-1}\) and \(F_{a}: \mathrm{K}^{\text{D}}_{I,n}\to\mathrm{K}^{\text{D}}_{I,n+1}\). For an irreducible \(M \in \mathrm{K}^{\text{D}}_{I,n}\) and \(a\in I\), define \(\tilde{e}_{a} M \in \mathrm{K}^{\text{D}}_{I,n-1}\) to be the socle of \(E_{a}M\). Define \(\tilde{f}_{a} M \in \mathrm{K}^{\text{D}}_{I,n+1}\) to be the cosocle of \(F_{a}M\). In fact, \(\tilde{f}_{a}M\) is always irreducible, and \(\tilde{e}_{a}M\) is a zero module or irreducible. The ring \(\mathrm{H}^{\text{D}}_{0}=\mathbf{C}\oplus\mathbf{C}\) has two irreducible modules \(\phi_{\pm}\). We understand \beginalign* E_a((b)) & = \tildee_a((b))= \begincases \phi_\pm & if \(a=b^{\pm1}\), 0 & otherwise, \endcases F_a(\phi_\pm) & =\tildef_a(\phi_\pm)=(a^\pm1). \endalign* Let \(V_{\theta}\) be as in Proposition 2.2. \beginconj \beginenumerate
\(\mathrm{K}^{\text{D}}\) is isomorphic to \((V_{\theta})_{\mathbf{A}}/(q-1)(V_{\theta})_{\mathbf{A}}\).
\(V_{\theta}\) has a crystal basis and an upper global basis. \label_i_
The elements of \(\mathrm{K}^{\text{D}}_{I}\) associated to irreducible representations correspond to the upper global basis of \(V_{\theta}\) at \(q=1\).
The operators \(\tilde{F}_{i}\) and \(\tilde{E}_{i}\) correspond to \(\tilde{f}_{i}\) and \(\tilde{e}_{i}\), respectively. \endenumerate \endconj Consider \(\tilde{H} = \mathrm{H}^{\text{D}}_{n} \otimes \mathbf{C}[\theta]/(\theta^{2}-1)\) with multiplication \(\theta T_{1}=T_{0}\theta\), \(X_{1}\theta = \theta X_{1}^{-1}\) and \(\theta\) commuting with all other generators. Then \(\tilde{H}\) is isomorphic to the specialization of the affine Hecke algebra of type B in which the generator for the node corresponding to the short root has eigenvalues \(\pm 1\). This explains why the crystal graph in the above case is a double covering of the crystal graph for the same \(\mathbf{Z}_{2}\times\mathbf{Z}\)-orbit in type \(B\). (See Remark 2.5 (v).) \section*Acknowledgment. The first author is partially supported by Grant-in-Aid for Scientific Research (B) 18340007, Japan Society for the Promotion of Science. \beginthebibliography99
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