Classification of graded Hecke algebras for complex reflection groups

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

Last update: 22 January 2014

The graded Hecke algebras $H_{\text{gr}}$

In [Lus1989], Lusztig gives a definition of graded Hecke algebras for real reflection groups which is different from the definition in Section 1, which applies to more general groups. It is not obvious that Lusztig’s algebras are examples of the graded Hecke algebras defined in Section 1. In this section, we show explicitly how the definition of Section 1 includes Lusztig’s algebras.

Let $W$ be a finite real reflection group acting on $V$ and let $R$ be the root system of $W\text{.}$ Let ${\alpha }_1,\dots ,{\alpha }_n$ be a choice of simple roots in $R$ and let $s_1,\dots ,s_n$ be the corresponding simple reflections in $W\text{.}$ Let $s_{\alpha }$ be the reflection in the root $\alpha$ so that, for $v\in V,$ $$s_{\alpha }v=v-⟨v,{\alpha }^{\vee }⟩\alpha ,\text{where}\hspace{0.17em}{\alpha }^{\vee }=2\alpha /⟨\alpha ,\alpha ⟩\text{.}$$ Let $R^{+}=\{\alpha >0\}$ denote the set of positive roots in $R\text{.}$

Let $k_{\alpha }$ be fixed complex numbers indexed by the roots $\alpha \in R$ satisfying $$\begin{array}{cc}{k}_{w\alpha }=k_{\alpha },\text{for all}\hspace{0.17em}w\in W,\hspace{0.17em}\alpha \in R\text{.}& \text{(3.1)}\end{array}$$ This amounts to a choice of either one or two “parameters”, depending on whether all roots in $R$ are the same length or not. As in Section 1, let $ℂW=ℂ\text{-span}\left\{t_g\hspace{0.17em}\right|\hspace{0.17em}g\in W\},$ with $t_g{t}_h=t_{gh},$ and let $S\left(V\right)$ be the symmetric algebra of $V\text{.}$ Lusztig [Lus1989] defines the “graded Hecke algebra” with parameters $\left\{k_{\alpha }\right\}$ to be the unique algebra structure $H_{\text{gr}}$ on the vector space $S\left(V\right)\otimes ℂW$ such that

(3.2 a)	$S\left(V\right)=S\left(V\right)\otimes 1$ is a subalgebra of $H_{\text{gr}},$
(3.2 b)	$ℂW=1\otimes ℂW$ is a subalgebra of $H_{\text{gr}},$ and
(3.2 c)	$t_{s_i}v=\left(s_{i}v\right)t_{s_i}-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩,$ for all $v\in V$ and simple reflections $s_i$ in the simple roots ${\alpha }_i\text{.}$

We shall show that every algebra $H_{\text{gr}}$ as defined by (3.2 a-c) is a graded Hecke algebra $A$ for a specific set of skew symmetric bilinear forms $a_g\text{.}$

Let $k_{\alpha }\in ℂ$ as in (3.1). Use the notation $$\begin{array}{cc}h=\frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }{\alpha }^{\vee }{t}_{s_{\alpha }},\text{so that}⟨v,h⟩=\frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }⟨v,{\alpha }^{\vee }⟩t_{s_{\alpha }}& \text{(3.3)}\end{array}$$ for $v\in V\text{.}$ The element $h$ should be viewed as an element of $V\otimes ℂW,$ and $⟨v,h⟩\in ℂW\text{.}$ With this notation, let $A$ be the algebra (as in Section 1) generated by $V$ and $ℂW$ with relations $$\begin{array}{cc}{t}_{g}v=\left(gv\right)t_g\text{and}[v,w]=-[⟨v,h⟩,⟨w,h⟩],\text{for}\hspace{0.17em}v,w\in V,\hspace{0.17em}g\in W\text{.}& \text{(3.4)}\end{array}$$ Note that $A$ is defined by the bilinear forms $$a_g(v,w)=\frac{1}{4}\sum _{\underset{g=s_{\alpha }{s}_{\beta }}{\alpha ,\beta >0}}{k}_{\alpha }{k}_{\beta }(⟨v,{\beta }^{\vee }⟩⟨w,{\alpha }^{\vee }⟩-⟨v,{\alpha }^{\vee }⟩⟨w,{\beta }^{\vee }⟩)\text{.}$$

The following theorem shows that the algebra $A$ satisfies the defining conditions (3.2 a-c) of the algebra $H_{\text{gr}}\text{.}$

Let $W$ be a finite real reflection group and let $A$ be the algebra defined by (3.4).

(a)	As vector spaces, $A\cong S\left(V\right)\otimes ℂW$ (and hence, $A$ is a graded Hecke algebra).
(b)	If $\stackrel{\sim }{v}=v-⟨v,h⟩$ for $v\in V,$ then $$[\stackrel{\sim }{v},\stackrel{\sim }{w}]=0\text{and}{t}_{s_i}\stackrel{\sim }{v}=\left(\widetilde{s_{i}v}\right)t_{s_i}-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩,$$ for all $v,w\in V$ and simple reflections $s_i$ in $W\text{.}$

		Proof.
	First note that if $u,v\in V$ then $$\begin{array}{cc}[u,⟨v,h⟩]=\frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }⟨v,{\alpha }^{\vee }⟩⟨u,{\alpha }^{\vee }⟩\alpha t_{s_{\alpha }}=[v,⟨u,h⟩]\text{.}& \text{(}*\text{)}\end{array}$$ Thus, for $u,v,w\in V,$ $$\begin{array}{cc}\begin{array}{ccc}[u,[v,w]]+[w,[u,v]]+[v,[w,u]]& =& [u,[⟨w,h⟩,⟨v,h⟩]]+[w,[⟨v,h⟩,⟨u,h⟩]]+[v,[⟨u,h⟩,⟨w,h⟩]]\\ & =& [[u,⟨w,h⟩],⟨v,h⟩]+[⟨w,h⟩,[u,⟨v,h⟩]]+[[w,⟨v,h⟩],⟨u,h⟩]\\ & & +[⟨v,h⟩,[w,⟨u,h⟩]]+[[v,⟨u,h⟩],⟨w,h⟩]+[⟨u,h⟩,[v,⟨w,h⟩]]\\ & =& [[w,⟨u,h⟩],⟨v,h⟩]+[⟨w,h⟩,[v,⟨u,h⟩]]+[[v,⟨w,h⟩],⟨u,h⟩]\\ & & +[⟨v,h⟩,[w,⟨u,h⟩]]+[[v,⟨u,h⟩],⟨w,h⟩]+[⟨u,h⟩,[v,⟨w,h⟩]]\\ & =& 0\text{.}\end{array}& \text{(3.6)}\end{array}$$ For $v\in V,$ $h\in W,$ and $s_i$ a simple reflection, $$\begin{array}{cc}\begin{array}{ccc}{t}_{s_i}⟨v,h⟩t_{s_i}& =& \frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }⟨v,{\alpha }^{\vee }⟩t_{s_{s_i\alpha }}=\left(\frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }⟨v,s_i{\alpha }^{\vee }⟩t_{s_{\alpha }}\right)+k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i}\\ & =& \left(\frac{1}{2}\sum _{\alpha >0}{k}_{\alpha }⟨s_{i}v,{\alpha }^{\vee }⟩t_{s_{\alpha }}\right)+k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i}=⟨s_{i}v,h⟩+k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i}\text{.}\end{array}& \text{(3.7)}\end{array}$$ Using this equality, we obtain $$\begin{array}{cc}\begin{array}{ccc}{t}_{s_i}[v,w]t_{s_i}& =& -t_{s_i}[⟨v,h⟩,⟨w,h⟩]t_{s_i}\\ & =& -[⟨s_{i}v,h⟩+k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i},⟨s_{i}w,h⟩+k_{{\alpha }_i}⟨w,{\alpha }_i^{\vee }⟩t_{s_i}]\\ & =& [s_{i}v,s_{i}w]-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩[t_{s_i},⟨s_{i}w,h⟩]-k_{{\alpha }_i}⟨w,{\alpha }_i^{\vee }⟩[⟨s_{i}v,h⟩,t_{s_i}]\\ & =& [s_{i}v,s_{i}w]-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩(t_{s_i}⟨s_{i}w,h⟩t_{s_i}-⟨s_{i}w,h⟩)t_{s_i}\\ & & +k_{{\alpha }_i}⟨w,{\alpha }_i^{\vee }⟩(t_{s_i}⟨s_{i}v,h⟩t_{s_i}-⟨s_{i}v,h⟩)t_{s_i}\\ & =& [s_{i}v,s_{i}w]-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩(⟨w,h⟩t_{s_i}+k_{{\alpha }_i}⟨s_{i}w,{\alpha }_i^{\vee }⟩-⟨s_{i}w,h⟩t_{s_i})\\ & & +k_{{\alpha }_i}⟨w,{\alpha }_i^{\vee }⟩(⟨v,h⟩t_{s_i}+k_{{\alpha }_i}⟨s_{i}v,{\alpha }_i^{\vee }⟩-⟨s_{i}v,h⟩t_{s_i})\\ & =& [s_{i}v,s_{i}w]-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩⟨w,{\alpha }_i^{\vee }⟩⟨{\alpha }_i,h⟩t_{s_i}-k_{{\alpha }_i}^2⟨v,{\alpha }_i^{\vee }⟩⟨w,s_i{\alpha }_i^{\vee }⟩\\ & & +k_{{\alpha }_i}⟨w,{\alpha }_i^{\vee }⟩⟨v,{\alpha }_i^{\vee }⟩⟨{\alpha }_i,h⟩t_{s_i}+k_{{\alpha }_i}^2⟨w,{\alpha }_i^{\vee }⟩⟨v,s_i{\alpha }_i^{\vee }⟩\\ & =& [s_{i}v,s_{i}w]\text{.}\end{array}& \text{(3.8)}\end{array}$$ The two identities (3.6) and (3.8), as in (1.3) and (1.4), show that the algebra $A$ is isomorphic to $S\left(V\right)\otimes ℂW\text{.}$ (b) This can now be proved by direct computation. If $v,w\in V$ then $$[\stackrel{\sim }{v},\stackrel{\sim }{w}]=[v-⟨v,h⟩,w-⟨w,h⟩]=[v,w]+[⟨v,h⟩,⟨w,h⟩]-[v,⟨w,h⟩]+[w,⟨v,h⟩]=0,$$ by equation (3.4) and equation $\text{(}*\text{)}$ in the proof of Theorem 3.5. If $v\in V$ and $s_i$ is a simple reflection then, by (3.7), $$t_{s_i}\stackrel{\sim }{v}{t}_{s_i}=t_{s_i}v{t}_{s_i}-t_{s_i}⟨v,h⟩t_{s_i}=s_{i}v-⟨s_{i}v,h⟩-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i}=\widetilde{s_{i}v}-k_{{\alpha }_i}⟨v,{\alpha }_i^{\vee }⟩t_{s_i}\text{.}$$ $\square$

Theorem 3.5b shows that if $A$ is the graded Hecke algebra defined by (3.4), then the elements $\stackrel{\sim }{v},$ for $v\in V,$ generate a subalgebra of $A$ isomorphic to $S\left(V\right)$ and these elements together with the $t_{s_i}$ satisfy the relations of (3.2 c). Since part (a) of Theorem 3.5 shows that $A$ is isomorphic to $S\left(V\right)\otimes ℂW$ as a vector space, it follows that $A$ satisfies the conditions (3.2 a-c), relations which uniquely define the graded Hecke algebra $A\text{.}$ Thus, Lusztig’s algebras are special cases of the graded Hecke algebras defined in Section 1. Furthermore, by comparing the dimensions of the parameter spaces, we see that there are graded Hecke algebras that are not isomorphic to algebras defined by Lusztig for the Coxeter groups $F_4,$ $H_3,$ $H_4,$ and $I_2\left(m\right)\text{.}$

Notes and references

This is a typed version of Classification of graded Hecke algebras for complex reflection groups by Arun Ram and Anne V. Shepler.

Research of the first author supported in part by the National Security Agency and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. Research of the second author supported in part by National Science Foundation grant DMS-9971099.

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