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

Introduction

This paper is motivated by a general effort to generalize the theory of Weyl groups and their relation to groups of Lie type to the setting of complex reflection groups. One natural question is whether there are affine Hecke algebras corresponding to complex reflection groups. If they exist then it might be possible to use these algebras to build an analogue of the Springer correspondence for complex reflection groups.

A priori, one knows how to construct affine Hecke algebras corresponding only to Weyl groups since both a finite real reflection group $W$ and a $W\text{-invariant}$ lattice (the existence of which forces $W$ to be a Weyl group) are needed in the construction. Our search for analogues of graded Hecke algebras for complex reflection groups was motivated by Lusztig’s results [Lus1988] showing that the geometric information contained in the affine Hecke algebra can be recovered from the corresponding graded Hecke algebra. Lusztig [Lus1989] defines the graded Hecke algebra for a finite Weyl group $W$ with reflection representation $V\text{.}$ Let $t_g,$ $g\in W,$ be a basis for the group algebra $ℂW$ of $W$ and let $k_{\alpha }\in ℂ$ be “parameters” indexed by the roots in the root system of $W$ such that $k_{\alpha }$ depends only on the length of the root $\alpha \text{.}$ Then the graded Hecke algebra $H_{\text{gr}}$ depending on the parameters $k_{\alpha }$ is the (unique) algebra structure on $S\left(V\right)\otimes ℂW$ such that

(a)	the symmetric algebra of $V,$ $S\left(V\right)=S\left(V\right)\otimes 1,$ is a subalgebra of $H_{\text{gr}},$
(b)	the group algebra $ℂW=1\otimes ℂW=\text{span-}\{1\otimes t_g\hspace{0.17em}|\hspace{0.17em}g\in W\}$ is a subalgebra of $H_{\text{gr}},$ and
(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 ${\alpha }_i\text{.}$

This definition applies to all finite real reflection groups $W$ since the simple roots and simple reflections are well defined. Unfortunately, the need for simple reflections in the construction makes it unclear how to define analogues for complex reflection groups.

For finite real reflection groups, the graded Hecke algebra $H_{\text{gr}}$ is a “semidirect product” of the polynomial ring $S\left(V\right)$ and the group algebra $ℂW\text{.}$ Drinfeld [Dri1986] defines a different type of semidirect product of $S\left(V\right)$ and $ℂW,$ and Drinfeld’s construction applies to all finite subgroups $G$ of $GL\left(V\right)\text{.}$ In this paper, we

(1)	Classify all the algebras obtained by applying Drinfeld’s construction to finite complex reflection groups $G,$
(2)	Show that every graded Hecke algebra $H_{\text{gr}}$ (as defined by Lusztig) for a finite real reflection group is isomorphic to an algebra obtained by Drinfeld’s construction by giving explicit isomorphisms between these algebras.

The results from (2) show how Drinfeld’s construction is a true generalization of Lusztig’s construction of graded Hecke algebras, something which is not obvious. Our classification in (1) gives a complete solution to the problem of finding all graded Hecke algebras for finite reflection groups.

A consequence of our classification is that there exist graded Hecke algebras for finite real reflection groups which are not obtained with Lusztig’s construction. In this sense, Drinfeld’s construction is a strict generalization of the algebras $H_{\text{gr}}\text{.}$ These new algebras, and the algebras corresponding to complex reflection groups that are not real reflection groups, deserve further study and probably have interesting representation theories.

For us, one surprising result of our classification is that no nontrivial graded Hecke algebra structures exist for many complex reflection groups. In some sense, this is disappointing, as we would have liked to find nontrivial and interesting structures for each complex reflection group.

It might be that we have not yet hit upon the “right” definition of graded Hecke algebras. For example, we show that there do not exist nontrivial graded Hecke algebra structures, according to Drinfeld’s definition, for any of the complex reflection groups $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$ when $r>2$ and $n>3\text{.}$ On the other hand, in the last section of this paper we are able to construct algebras that “look” like they ought to be graded Hecke algebras corresponding to these groups. Is it possible that there is a “better” definition of graded Hecke algebras which applies to complex reflection groups and which includes the algebras that we introduce in Section 5 as examples?

Acknowledgements. We thank C. Kriloff for numerous stimulating conversations during our work on this paper. A. Ram thanks the Newton Institute for the Mathematical Sciences at Cambridge University for hospitality and support (EPSRC Grant No. GR K99015) during Spring 2001 when the writing of this paper was completed.

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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