English

Endoscopy for Hecke categories, character sheaves and representations

Representation Theory 2021-08-24 v3 Algebraic Geometry

Abstract

For a split reductive group GG 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 HH with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on GG with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group HH, after passing to asymptotic versions. The other is a similar relationship between representations of G(Fq)G(\mathbb{F}_q) with a fixed semisimple parameter and unipotent representations of H(Fq)H(\mathbb{F}_{q}).

Keywords

Cite

@article{arxiv.1904.01176,
  title  = {Endoscopy for Hecke categories, character sheaves and representations},
  author = {George Lusztig and Zhiwei Yun},
  journal= {arXiv preprint arXiv:1904.01176},
  year   = {2021}
}

Comments

This is a revised version of the paper with the same title published in Forum Math. Pi 8 (2020), e12. An error on a 3-cocycle is corrected and main statements are simplified. A Corrigendum follows the main article. 57 pages

R2 v1 2026-06-23T08:26:16.187Z