https://orcid.org/0000-0001-6516-6739
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Maarten Solleveld, Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone, The Quarterly Journal of Mathematics, Volume 76, Issue 1, March 2025, Pages 109–146, https://doi.org/10.1093/qmath/haae065
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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 |$G \times \mathbb{C}^\times$|-equivariant semisimple complex of sheaves on the nilpotent cone |$\mathfrak g_N$| in the Lie algebra of G.
From there we provide an algebraic description of the |$G \times \mathbb{C}^\times$|-equivariant bounded derived category of constructible sheaves on |$\mathfrak g_N$|. 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.
This paper prepares for a study of representations of reductive p-adic groups with a fixed infinitesimal central character. In sequel papers [34, 35], that will lead to proofs of the generalized injectivity conjecture and of the Kazhdan–Lusztig conjecture for p-adic groups.
