Hochschild cohomology and graded Hecke algebras

Shepler, Anne; Witherspoon, Sarah

doi:10.1090/S0002-9947-08-04396-1

Hochschild cohomology and graded Hecke algebras
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by Anne V. Shepler and Sarah Witherspoon PDF

Trans. Amer. Math. Soc. 360 (2008), 3975-4005 Request permission

Abstract:

We develop and collect techniques for determining Hochschild cohomology of skew group algebras $S(V)\# G$ and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups $G(r,p,n)$ to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of $S(V)\# G$ and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of $G$ on $V$, and we show that there exist nontrivial graded Hecke algebras for $G(r,1,n)$, in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

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