ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

Abstract

© 2020 Journal of Materials Research. All rights reserved. For a reductive group 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 with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group, after passing to asymptotic versions. The other is a similar relationship between representations of with a fixed semisimple parameter and unipotent representations of.