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Hochschild cohomology and graded Hecke algebras

Author(s): Anne V. Shepler; Sarah Witherspoon
Journal: Trans. Amer. Math. Soc. 360 (2008), 3975-4005.
MSC (2000): Primary 16E40, 16S80
Posted: March 20, 2008
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Abstract: We develop and collect techniques for determining Hochschild cohomology of skew group algebras $ S(V)\char93 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)\char93 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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Additional Information:

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: ashepler@unt.edu

Sarah Witherspoon
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: sjw@math.tamu.edu

DOI: 10.1090/S0002-9947-08-04396-1
PII: S 0002-9947(08)04396-1
Keywords: Graded Hecke algebra, degenerate affine Hecke algebra, deformation, Hochschild cohomology, reflection group, hyperplane arrangement, Ariki-Koike algebra
Received by editor(s): January 20, 2006
Posted: March 20, 2008
Additional Notes: The first author was partially supported by NSF grant \#DMS-0402819
The second author was partially supported by NSF grant \#DMS-0443476 and the Alexander von Humboldt Foundation
Dedicated: We dedicate this article to Sergey Yuzvinsky on the occasion of his 70th birthday.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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