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arXiv · 2205.07490

Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone

Maarten Solleveld
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Abstract

Graded Hecke algebras can be constructed geometrically, with constructible sheaves and equivariant cohomology. The input consists of a complex reductive group G (possibly disconnected) and a cuspidal local system on a nilpotent orbit for a Levi subgroup of G. We prove that every such "geometric" graded Hecke algebra is naturally isomorphic to the endomorphism algebra of a certain G x C*-equivariant semisimple complex of sheaves on the nilpotent cone $g_N$. From there we provide an algebraic description of the G x C*-equivariant bounded derived category of constructible sheaves on $g_N$. Namely, it is equivalent with the bounded derived category of finitely generated differential graded modules of a suitable direct sum of graded Hecke algebras. This can be regarded as a categorification of graded Hecke algebras.

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Maarten Solleveld. 2022-05-16. Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone. https://arxiv.org/abs/2205.07490

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