Classification of graded Hecke algebras for complex reflection groups

Abstract

Abstract. The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V ). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Mathematics Subject Classification (2000). 20C, 20F, 52B.Keywords. Reflection group, Coxeter group, Weyl group, affine Hecke algebra, Iwahori-Hecke algebra, representation theory, graded Hecke algebra. IntroductionThis 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 -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 [Lu2] showing that the geometric information contained in the affine Hecke algebra can be recovered from the corresponding graded Hecke algebra. Lusztig [Lu] defines the graded Hecke algebra for a finite Weyl group W with reflection representation V . Let t g , g ∈ W , be a basis for the group algebra CW of W and let k α ∈ C be "parameters" indexed by the roots in the root system of W such that k α depends only on the length of the root α. Then the graded Hecke algebra H gr depending on the parameters k α is the (unique) algebra structure on S(V ) ⊗ CW such that (a) the symmetric algebra of V ,for all v ∈ V and simple reflections s i in the simple roots α i . 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 gr is a "semidirect product" of the polynomial ring S(V ) and the group algebra CW . Drinfeld [Dr] defines a different type of semidirect product of S(V ) and CW , and Drinfeld's construction applies to all finite subgroups G of GL(V ). In this paper, we(1) classify all the algebras obtained by applying Dr...

Hochschild cohomology and graded Hecke algebras

“…We characterize all possible graded Hecke algebras, both in relation to Hochschild cohomology (Theorem 8.7) and in relation to defining skew-symmetric forms (Corollary 8.17). Corollary 8.17 generalizes a result of Ram and the first author [22], first formulated by Drinfeld [9] for Coxeter groups.…”

“…As a consequence of Theorems 5.1 and 8.7, if G = G(r, p, n) with r ≥ 3, n ≥ 4, and V is its natural reflection representation, then there are no nontrivial graded Hecke algebras: The relevant Hochschild cohomology, while nonzero, is not of the required form. This nonexistence of a graded Hecke algebra was discovered by Ram and the first author [22], and now we may view their result in the context of algebraic deformation theory and Hochschild cohomology. This negative result inspired their ad hoc construction of a "different graded Hecke algebra" for G(r, 1, n), which coincides with an algebra defined by Dezélée [7] in case r = 2.…”

“…Our results in case (1) above show that the lack of nontrivial graded Hecke algebras (when r ≥ 3, n ≥ 4, see [22]) stems from a lack of certain types of elements in HH 2 (S(V )#G). This conclusion motivates our computation in case (2), leading to the aforementioned nontrivial graded Hecke algebras under a nonfaithful representation of G.…”

“…We construct a natural graded Hecke algebra for G(r, 1, n) using a nonfaithful representation, and we show that this construction coincides with the algebra of Ram and the first author [22] (Theorem 9.8). We further give the generic form of a graded Hecke algebra associated to G(r, 1, n), depending on many parameters, in (9.1).…”

Dirac cohomology for symplectic reflection algebras

“…Proposition 2.2 ( [Dr], [RS,Theorem 1.9]). The algebra H has the PBW property if and only if the following properties hold simultaneously:…”