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...

Abstract. Braverman and Gaitsgory gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincaré-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincaré-Birkhoff-Witt property over (nonsemisimple) group algebras. We prove our main results by exploiting a double complex adapted from Guccione, Guccione, and Valqui (formed from a Koszul complex and a resolution of the group), giving a practical way to analyze Hochschild cohomology and deformations of skew group algebras in positive characteristic. We apply these conditions to graded Hecke algebras and Drinfeld orbifold algebras (including rational Cherednik algebras and symplectic reflection algebras) in arbitrary characteristic, with special interest in the case when the characteristic of the underlying field divides the order of the acting group.

We define Drinfeld orbifold algebras as filtered algebras deforming the skew group algebra (semidirect product) arising from the action of a finite group on a polynomial ring. They simultaneously generalize Weyl algebras, graded (or Drinfeld) Hecke algebras, rational Cherednik algebras, symplectic reflection algebras, and universal enveloping algebras of Lie algebras with group actions. We give necessary and sufficient conditions on defining parameters to obtain Drinfeld orbifold algebras in two general formats, both algebraic and homological. Our algebraic conditions hold over any field of characteristic other than two, including fields whose characteristic divides the order of the acting group. We explain the connection between Hochschild cohomology and a Poincaré-Birkhoff-Witt property explicitly (using Gerstenhaber brackets). We also classify those deformations of skew group algebras which arise as Drinfeld orbifold algebras and give applications for abelian groups.

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.

When a finite group acts linearly on a complex vector space, the natural semi-direct product of the group and the polynomial ring over the space forms a skew group algebra. This algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics. Its Hochschild cohomology predicts various Hecke algebras and deformations of the orbifold. In this article, we investigate the ring structure of the Hochschild cohomology of the skew group algebra. We show that the cup product coincides with a natural smash product, transferring the cohomology of a group action into a group action on cohomology. We express the algebraic structure of Hochschild cohomology in terms of a partial order on the group (modulo the kernel of the action). This partial order arises after assigning to each group element the codimension of its fixed point space. We describe the algebraic structure for Coxeter groups, where this partial order is given by the reflection length function; a similar combinatorial description holds for an infinite family of complex reflection groups.Let K be the kernel of the representation of G acting on V :Definition 2.2. Define a binary relation ≤ on G by g ≤ h whenever By Lemma 2.1, this codimension condition holds exactly whenThis induces a binary relation ≤ on the quotient group G/K as well: For g, h in G, define gK ≤ hK when g ≤ h. Note the relation does not depend on choice of representatives of cosets, as V g = V h whenever gK = hK for g, h in G.The relation ≤ appears in work of Brady and Watt [4] on orthogonal transformations. Their arguments apply equally well to our setting of isometries with