Generic character sheaves on parahoric subgroups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911099489615872 |
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| author | Bezrukavnikov, Roman Chan, Charlotte |
| author_facet | Bezrukavnikov, Roman Chan, Charlotte |
| contents | We study parabolic induction producing $\ell$-adic sheaves on a parahoric subgroup scheme in the loop group of a reductive group. Under a genericity assumption on the input data, we prove that it produces conjugation equivariant perverse sheaves on the parahoric subgroup; this is upgraded to a $t$-exact equivalence of categories of $\ell$-adic sheaves. An iterative version of the construction produces such a perverse sheaf starting from a geometric analogue of the data considered by J.-K. Yu and J. Kim. We prove, under a mild condition on $q$, that generic parabolic induction from a parahoric torus realizes the character of the representation arising from the associated parahoric Deligne--Lusztig induction, which is known to parametrize the Fintzen--Kaletha--Spice twist of types. In the simplest interesting setting, our construction produces a simple perverse sheaf associated to a sufficiently nontrivial multiplicative local system on a torus, resolving a conjecture of Lusztig. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic character sheaves on parahoric subgroups Bezrukavnikov, Roman Chan, Charlotte Representation Theory Algebraic Geometry We study parabolic induction producing $\ell$-adic sheaves on a parahoric subgroup scheme in the loop group of a reductive group. Under a genericity assumption on the input data, we prove that it produces conjugation equivariant perverse sheaves on the parahoric subgroup; this is upgraded to a $t$-exact equivalence of categories of $\ell$-adic sheaves. An iterative version of the construction produces such a perverse sheaf starting from a geometric analogue of the data considered by J.-K. Yu and J. Kim. We prove, under a mild condition on $q$, that generic parabolic induction from a parahoric torus realizes the character of the representation arising from the associated parahoric Deligne--Lusztig induction, which is known to parametrize the Fintzen--Kaletha--Spice twist of types. In the simplest interesting setting, our construction produces a simple perverse sheaf associated to a sufficiently nontrivial multiplicative local system on a torus, resolving a conjecture of Lusztig. |
| title | Generic character sheaves on parahoric subgroups |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2401.07189 |