Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929678746386432 |
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| author | Solleveld, Maarten |
| author_facet | Solleveld, Maarten |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_07490 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone Solleveld, Maarten Algebraic Geometry Representation Theory 14F08, 20C08, 22E57 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. |
| title | Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone |
| topic | Algebraic Geometry Representation Theory 14F08, 20C08, 22E57 |
| url | https://arxiv.org/abs/2205.07490 |