Twisted graded Hecke algebras

Witherspoon, Sarah

doi:10.1016/j.jalgebra.2007.05.025

Cited by 13 publications

(18 citation statements)

References 22 publications

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“…These potentially different characters χ α g can lead to different semi-invariants. In this broader setting, deformations of S(V )# α G are treated in [5,6,26,27].…”

Section: Hochschild Cohomology and Graded Hecke Algebras 3981mentioning

confidence: 99%

“…Consider S(V )#G to be a graded algebra where elements of V have degree 1 and elements of G have degree 0. In [27], a graded Hecke algebra is defined over C[t]: It is a quotient of T (V )#G[t] by the ideal generated by all vw −wv − g∈G a g (v, w)tg, whose associated graded C[t]-algebra is isomorphic to S(V )#G [t]. To obtain our Definition 8.1, simply substitute any nonzero complex number for t.…”

Section: Is the Corresponding Linear Combination Of Elements Y(z)rpgsmentioning

confidence: 99%

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Hochschild cohomology and graded Hecke algebras

Shepler¹,

Witherspoon²

2008

Trans. Amer. Math. Soc.

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We dedicate this article to Sergey Yuzvinsky on the occasion of his 70th birthday.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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“…These potentially different characters χ α g can lead to different semi-invariants. In this broader setting, deformations of S(V )# α G are treated in [5,6,26,27].…”

Section: Hochschild Cohomology and Graded Hecke Algebras 3981mentioning

confidence: 99%

Section: Is the Corresponding Linear Combination Of Elements Y(z)rpgsmentioning

confidence: 99%

Hochschild cohomology and graded Hecke algebras

Shepler¹,

Witherspoon²

2008

Trans. Amer. Math. Soc.

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“…By [25,Thm 3.2], the resulting formal deformation of S(V )# α G becomes a (twisted) graded Hecke algebra [25] when the scalars are restricted to C [t]. In this case, that means the associated deformation of…”

Section: Deformations Of Group Crossed Productsmentioning

confidence: 99%

“…We use Hochschild cohomology to prove that the resulting deformations are nontrivial, by showing that their associated infinitesimals are not coboundaries. We apply the results of [25] to show how our deformation formula leads, in special cases, to (twisted) graded Hecke algebras (Example 4.13). Due to restrictive hypotheses, our formula does not give rise to very many of the examples of (twisted) graded Hecke algebras in [7,8,18,21,25].…”

Section: Introductionmentioning

confidence: 99%

“…We apply the results of [25] to show how our deformation formula leads, in special cases, to (twisted) graded Hecke algebras (Example 4.13). Due to restrictive hypotheses, our formula does not give rise to very many of the examples of (twisted) graded Hecke algebras in [7,8,18,21,25]. However it does give an infinite series of universal deformation formulas based on noncocommutative Hopf algebras, as well as algebras thereby undergoing nontrivial deformation.…”

Section: Introductionmentioning

confidence: 99%

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Skew Derivations and Deformations of a Family of Group Crossed Products

Witherspoon

2006

Communications in Algebra

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Abstract. We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. We show that the resulting deformations are nontrivial by a comparison with Hochschild cohomology. The universal deformation formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations, and are universal in the sense that they apply to deform all algebras with such Hopf algebra actions.

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Graded Hecke Algebras for Disconnected Reductive Groups

2018

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We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G.We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G, M, L) and they are closely related to Langlands parameters.Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of "logarithms" of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper. Erratum. Theorem 3.4 and Proposition 3.22 were not entirely correct as stated. This is repaired in a new appendix.

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Twisted graded Hecke algebras

Cited by 13 publications

References 22 publications

Hochschild cohomology and graded Hecke algebras

Hochschild cohomology and graded Hecke algebras

Skew Derivations and Deformations of a Family of Group Crossed Products

Graded Hecke Algebras for Disconnected Reductive Groups

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