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ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

Part of: Representation theory of groups Linear algebraic groups and related topics (Co)homology theory

Published online by Cambridge University Press:  28 May 2020

GEORGE LUSZTIG  and
ZHIWEI YUN
GEORGE LUSZTIG
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA; gyuri@math.mit.edu
ZHIWEI YUN
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA; zyun@mit.edu

Abstract

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For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

MSC classification

Primary: 20G40: Linear algebraic groups over finite fields
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 20C08: Hecke algebras and their representations 20C33: Representations of finite groups of Lie type
Type
Algebra
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e12
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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