Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone

2022/05/16 by Solleveld, Maarten
#14F08 #20C08 #22E57 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2205.07490

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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