Affine Hecke algebras of general type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 28 February 2013

Affine Hecke algebras of general type

Let $R$ be a reduced root system and let $R^{\vee }$ be the root system formed by the coroots ${\alpha }^{\vee }=2\alpha /⟨\alpha ,\alpha ⟩$ for $\alpha \in R\text{.}$ Let $W$ be the Weyl group of $R$ and fix a system of positive roots $R^{+}$ in $R\text{.}$ Let $\{{\alpha }_1,\dots ,{\alpha }_n\}$ be the corresponding simple roots and let $s_1,\dots ,s_n$ be the corresponding simple reflections in $W\text{.}$ The fundamental weights are defined by the equations $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$ and the lattices

$$P=\sum _{i=1}^n\mathbb{Z}{\omega }_i\text{and}Q=\sum _{i=1}^n\mathbb{Z}{\alpha }_i,$$

are the weight lattice and the root lattice, respectively. The Dynkin diagrams and the corresponding extended Dynkin diagrams are given in Figure 3. If $\Gamma$ is a Dynkin diagram or extended Dynkin diagram define

$$m_{ij}=\{\begin{array}{cc}2,& i j if ,\\ 3,& i j if ,\end{array}\text{and}{m}_{ij}=\{\begin{array}{cc}4,& i j if ,\\ 6,& i j if \text{.}\end{array}$$

4.1 Affine Weyl groups

The extended affine Weyl group is the group

$$\stackrel{\sim }{W}=W⋉P=\{w{t}_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}w\in W,\lambda \in P\},\text{where}w{t}_{\lambda }=t_{w\lambda }w,$$

for $w\in W$ and $\lambda \in P$ where $t_{\lambda }$ corresponds to translation by $\lambda \in P\text{.}$ Define $s_0\in \stackrel{\sim }{W}$ by the equation

$$\begin{array}{cc}{s}_0{s}_{{\varphi }^{\vee }}=t_{\varphi },\text{where}\hspace{0.17em}{\varphi }^{\vee }\hspace{0.17em}\text{is the highest root of}\hspace{0.17em}{R}^{\vee },& \text{(4.2)}\end{array}$$

see [Bou1968 Ch. IV §1 no. 2.1]. The subgroup $W_{\text{aff}}=W⋉Q$ of $\stackrel{\sim }{W}$ is presented by generators $s_0,s_1,\dots ,s_n$ and relations

$$s_i^2=1,0\le i\le n,\text{and}\underset{\underset{m_{ij}}{⏟}}{s_i{s}_j\dots }=\underset{\underset{m_{ij}}{⏟}}{s_j{s}_i\dots },i\ne j,$$

where the $m_{ij}$ are determined from the extended Dynkin diagram of the root system $R^{\vee }\text{.}$ Define

$$\begin{array}{cc}\Omega =\{g_i\hspace{0.17em}\mid \hspace{0.17em}{\omega }_i\hspace{0.17em}\text{is minuscule}\},\text{where}{g}_i{w}_0{w}_{0,i}=t_{{\omega }_i},& \text{(4.3)}\end{array}$$

$w_0$ is the longest element of $W$ and $w_{0,i}$ is the longest element of the group $⟨s_j\hspace{0.17em}\mid \hspace{0.17em}1\le j\le n,j\ne i⟩,$ see [Bou1968 Ch. IV §2 Prop. 6]. Then $\Omega \cong P/Q$ and each element $g\in \Omega$ corresponds to an automorphism of the extended Dynkin diagram of $R^{\vee },$ in the sense that

$$\begin{array}{cc}if$ g\in \Omega $\text{then}g{s}_i{g}^{-1}=s_{\sigma \left(i\right)},& \text{(4.4)}\end{array}$$

where $\sigma$ is the permutation of the nodes determined by the automorphism. Equation (4.4) means that $\stackrel{\sim }{W}=W_{\text{aff}}⋊\Omega \text{.}$ The usual length function on the Coxeter group $W_{\text{ff}}$ is extended to the group $\stackrel{\sim }{W}$ by

$$\ell \left(wg\right)=\ell \left(w\right),\text{for}\hspace{0.17em}w\in W_{\text{aff}}\hspace{0.17em}\text{and}\hspace{0.17em}g\in \Omega \text{.}$$

Let $L$ be a lattice such that $Q\subseteq L\subseteq P\text{.}$ View $L/Q$ as a subgroup of $\Omega \cong P/Q$ and let

$${\stackrel{\sim }{W}}_L=W⋉L=W_{\text{aff}}⋊(L/Q)\text{.}$$

Then $W_{\text{aff}}={\stackrel{\sim }{W}}_Q,$ $\stackrel{\sim }{W}={\stackrel{\sim }{W}}_P,$ and ${\stackrel{\sim }{W}}_L$ is a subgroup of $\stackrel{\sim }{W}\text{.}$

4.5 Affine braid groups

Let $L$ be a lattice such that $Q\subseteq L\subseteq P\text{.}$ The affine braid group ${\stackrel{\sim }{ℬ}}_L$ is the group given by generators $T_w,$ $w\in {\stackrel{\sim }{W}}_L,$ and relations

$$T_w{T}_{w\prime }=T_{ww\prime },\text{if}\hspace{0.17em}\ell \left(ww\prime \right)=\ell \left(w\right)+\ell \left(w\prime \right)\text{.}$$

Let ${\stackrel{\sim }{ℬ}}_{\text{aff}}={\stackrel{\sim }{ℬ}}_Q\text{.}$ View $L/Q$ as a subgroup of $\Omega \cong P/Q\text{.}$ Then ${ℬ}_L={ℬ}_{\text{aff}}⋊L/Q$ is presented by generators $T_i=T_{s_i},$ $0\le i\le n,$ and relations

$$\begin{array}{cc}\underset{\underset{m_{ij}}{⏟}}{T_i{T}_j\dots }=\underset{\underset{m_{ij}}{⏟}}{T_j{T}_i\dots },\text{and}g{T}_i{g}^{-1}=T_{\sigma \left(i\right)},\text{for}\hspace{0.17em}g\in \Omega ,& \text{(4.6)}\end{array}$$

where $\sigma$ is as in (4.4), and the $m_{ij}$ are specified by the extended Dynkin diagram of $R^{\vee }\text{.}$

Let $P^{+}={\sum }_{i=1}^n{\mathbb{Z}}_{\ge 0}{\omega }_i$ be the dominant weights in $P\text{.}$ Define elements $X^{\lambda },$ $\lambda \in P$ by

$$\begin{array}{cc}\begin{array}{ccc}{X}^{\lambda }& =& T_{t_{\lambda }},\hspace{0.17em}\text{if}\hspace{0.17em}\lambda \in P^{+},\hspace{0.17em}\text{and}\\ X^{\lambda }& =& X^{\mu }{\left(X^{\nu }\right)}^{-1},\text{if}\hspace{0.17em}\lambda =\mu -\nu \hspace{0.17em}\text{with}\hspace{0.17em}\mu ,\nu \in P^{+}\text{.}\end{array}& \text{(4.7)}\end{array}$$

By [Mac1423624, 3.4] and [Lus1989], the $X^{\lambda }$ are well defined and do not depend on the choice $\lambda =\mu -\nu ,$ and

$$\begin{array}{cc}{X}^{\lambda }{X}^{\mu }=X^{\mu }{X}^{\lambda }=X^{\lambda +\mu },\text{for}\hspace{0.17em}\lambda ,\mu \in P\text{.}& \text{(4.8)}\end{array}$$

Then $X^{\lambda }\in {\stackrel{\sim }{ℬ}}_L$ if and only if $\lambda \in L\text{.}$

4.9 Affine Hecke algebras

Fix $q\in {ℂ}^{*}\text{.}$ The affine Hecke algebra ${\stackrel{\sim }{H}}_L$ is the quotient of the group algebra $ℂ{\stackrel{\sim }{ℬ}}_L$ by the relations

$$\begin{array}{cc}(T_i-q)(T_i+q^{-1})=0,\text{for}\hspace{0.17em}0\le i\le n\text{.}& \text{(4.10)}\end{array}$$

In ${\stackrel{\sim }{H}}_L$ (see [Mac1995, 4.2]),

$$\begin{array}{cc}{X}^{\lambda }{T}_i=T_i{X}^{s_i\lambda }+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},\text{for}\hspace{0.17em}\lambda \in L,1\le i\le n\text{.}& \text{(4.11)}\end{array}$$

The Iwahori-Hecke algebra $H$ is the subalgebra of $\stackrel{\sim }{H}$ generated by $T_1,\dots ,T_n\text{.}$

To our knowledge, the following theorem is due to Bernstein and Zelevinsky in type A, and to Bernstein in general type (unpublished). Lusztig has given an exposition in [Lus1989]. We give a new proof which we believe is more elementary and more direct.

Theorem 4.12. (Bernstein, Zelevinsky, Lusztig [Lus1989]) Let $L$ be a lattice such that $Q\subseteq L\subseteq P,$ where $Q$ is the root lattice and $P$ is the weight lattice of the root system $R\text{.}$ Let $\stackrel{\sim }{H}={\stackrel{\sim }{H}}_L$ be the affine Hecke algebra corresponding to $L$ and let $ℂ\left[X\right]=\text{span}\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L\}\text{.}$ Let $W$ be the Weyl group of $R\text{.}$ Then the center of $\stackrel{\sim }{H}$ is

$$Z\left(\stackrel{\sim }{H}\right)=ℂ{\left[X\right]}^W=\{f\in ℂ\left[X\right]\hspace{0.17em}\mid \hspace{0.17em}wf=f\hspace{0.17em}\text{for every}\hspace{0.17em}w\in W\}\text{.}$$

		Proof.
	Assume $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w\in Z\left(\stackrel{\sim }{H}\right)\text{.}$$ Let $m\in W$ be maximal in Bruhat order subject to $c_{\gamma ,m}\ne 0$ for some $\gamma \in L\text{.}$ If $m\ne 1$ there exists a dominant $\mu \in L$ such that $c_{\gamma +\mu -m\mu ,m}=0$ (otherwise $c_{\gamma +\mu -m\mu ,m}\ne 0$ for every dominant $\mu \in L,$ which is impossible since $z$ is a finite linear combination of $X^{\lambda }{T}_w\text{).}$ Since $z\in Z\left(\stackrel{\sim }{H}\right)$ we have $$z=X^{-\mu }z{X}^{\mu }=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda -\mu }{T}_w{X}^{\mu }\text{.}$$ Repeated use of the relation (4.11) yields $$T_w{X}^{\mu }=\sum _{\nu \in L,v\in W}{d}_{\nu ,v}{X}^{\nu }{T}_v$$ where $d_{\nu ,v}$ are constants such that $d_{w\mu ,w}=1,$ $d_{\nu ,w}=0$ for $\nu \ne w\mu ,$ and $d_{\nu ,v}=0$ unless $v\le w\text{.}$ So $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w=\sum _{\lambda \in L,w\in W}\sum _{\nu \in L,v\in W}{c}_{\lambda ,w}{d}_{\nu ,v}{X}^{\lambda -\mu +\nu }{T}_v$$ and comparing the coefficients of $X^{\gamma }{T}_w$ gives $c_{\gamma ,m}=c_{\gamma +\mu -m\mu ,m}{d}_{m\mu ,m}\text{.}$ Since $c_{\gamma +\mu -m\mu ,m}=0$ it follows that $c_{\gamma ,m}=0,$ which is a contradiction. Hence $z={\sum }_{\lambda \in L}{c}_{\lambda }{X}^{\lambda }\in ℂ\left[X\right]\text{.}$ The relation (4.11) gives $$z{T}_i=T_{i}z=\left(s_{i}z\right)T_i+(q-q^{-1})z\prime$$ where $z\prime \in ℂ\left[X\right]\text{.}$ Comparing coefficients of $X^{\lambda }$ on both sides yields $z\prime =0\text{.}$ Hence $z{T}_i=\left(s_{i}z\right)T_i,$ and therefore $z=s_{i}z$ for $1\le i\le n\text{.}$ So $z\in ℂ{\left[X\right]}^W\text{.}$ $\square$

4.13 Deducing the ${\stackrel{\sim }{H}}_L$ representation theory from ${\stackrel{\sim }{H}}_P$

Although we have not taken this point of view in our presentation, the affine Hecke algebras defined above are naturally associated to a reductive algebraic group $G$ over $ℂ$ [KLu0862716] or a $p\text{-adic}$ Chevalley group [IMa1965]. In this formulation, the lattice $L$ is determined by the group of characters of the maximal torus of $G\text{.}$ It is often convenient to work only with the adjoint version or only with the simply connected version of the group $G$ and therefore it seems desirable to be able to derive the representation theory of the affine Hecke algebras ${\stackrel{\sim }{H}}_L$ from the representation theory of the affine Hecke algebra ${\stackrel{\sim }{H}}_P\text{.}$ The following theorem shows that this can be done in a simple way by using the extension of Clifford theory in the Appendix. In particular, Theorem A.13, can be used to construct all of the simple ${\stackrel{\sim }{H}}_L\text{-modules}$ from the simple ${\stackrel{\sim }{H}}_P\text{-modules.}$

Theorem 4.14. Let $L$ be a lattice such that $Q\subseteq L\subseteq P,$ where $Q$ is the root lattice and $P$ is the weight lattice of the root system $R\text{.}$ Let $\stackrel{\sim }{H}={\stackrel{\sim }{H}}_L$ be the affine Hecke algebra corresponding to $L\text{.}$ Then there is an action of a finite group $K$ on ${\stackrel{\sim }{H}}_P,$ acting by automorphisms, such that

$${\stackrel{\sim }{H}}_L={\left({\stackrel{\sim }{H}}_P\right)}^K,$$

is the subalgebra of fixed points under the action of the group $K\text{.}$

		Proof.
	There are two cases to consider, depending on whether the group $\Omega \cong P/Q$ is cyclic or not. $$\begin{array}{ccccccccccc}\text{Type}& A_{n-1}& B_n& C_n& D_{2n-1}& D_{2n}& E_6& E_7& E_8& F_4& G_2\\ \Omega & \mathbb{Z}/n\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}& \mathbb{Z}/4\mathbb{Z}& \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}& \mathbb{Z}/3\mathbb{Z}& \mathbb{Z}/4\mathbb{Z}& 1& 1& 1\end{array}$$ In each case we construct the group $L$ and its action on ${\stackrel{\sim }{H}}_P$ explicitly. This is necessary for the effective application of Theorem A.13 on examples. Case 1. If $\Omega$ is a cyclic group $\Omega$ then the subgroup $L/Q$ is a cyclic subgroup. Suppose $$\Omega =\{1,g,\dots ,g^{r-1}\}\text{and}L/Q=\{1,g^d,\dots ,g^{d(p-1)}\},$$ where $pd=r\text{.}$ Let $\zeta$ be a primitive $p\text{th}$ root of unity and define an automorphism $$\begin{array}{cccc}\sigma :& {\stackrel{\sim }{H}}_P& ⟶& {\stackrel{\sim }{H}}_P\\ & g& ⟼& \zeta g,\\ & T_i& ⟼& T_i,& 0\le i\le n\text{.}\end{array}$$ The map $\sigma$ is an algebra isomorphism since it preserves the relations in (4.6) and (4.10). Furthermore, $\sigma$ gives rise to a $\mathbb{Z}/p\mathbb{Z}$ action on ${\stackrel{\sim }{H}}_P$ and $$\begin{array}{cc}{\stackrel{\sim }{H}}_L={\left({\stackrel{\sim }{H}}_P\right)}^{\mathbb{Z}/p\mathbb{Z}}\text{.}& \text{(4.15)}\end{array}$$ Case 2. If the root system $R^{\vee }$ is of type $D_n,n$ even, then $\Omega \cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and the subgroups of $\Omega$ correspond to the intermediate lattices $Q\subseteq L\subseteq P\text{.}$ Suppose $$\Omega =\{1,g_1,g_2,g_1{g}_2\hspace{0.17em}\mid \hspace{0.17em}{g}_1^2=g_2^2,g_1{g}_2=g_2{g}_1\}\text{.}$$ and define automorphisms of ${\stackrel{\sim }{H}}_P$ by $$\begin{array}{cccc}{\sigma }_1:& {\stackrel{\sim }{H}}_P& ⟶& {\stackrel{\sim }{H}}_P\\ & g_1& ⟼& -g_1,\\ & g_2& ⟼& g_2,\\ & T_i& ⟼& T_i,\end{array}\text{and}\begin{array}{cccc}{\sigma }_2:& {\stackrel{\sim }{H}}_P& ⟶& {\stackrel{\sim }{H}}_P\\ & g_1& ⟼& g_1,\\ & g_2& ⟼& -g_2,\\ & T_i& ⟼& T_i\text{.}\end{array}$$ Then $$\begin{array}{cc}{\stackrel{\sim }{H}}_{L_1}={\left({\stackrel{\sim }{H}}_P\right)}^{{\sigma }_1},{\stackrel{\sim }{H}}_{L_2}={\left({\stackrel{\sim }{H}}_P\right)}^{{\sigma }_2},\text{and}{\stackrel{\sim }{H}}_Q={\left({\stackrel{\sim }{H}}_P\right)}^{⟨{\sigma }_1,{\sigma }_2⟩},& \text{(4.16)}\end{array}$$ where $L_1$ and $L_2$ are the two intermediate lattices strictly between $Q$ and $P\text{.}$ $\square$

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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