# Endoscopy for Hecke categories, character sheaves and representations

## TL;DR

This paper establishes an equivalence between certain categories related to a split reductive group over a finite field and its endoscopic group, linking character sheaves and representations with fixed parameters.

## Contribution

It proves a monoidal equivalence between blocks of the mixed Hecke category of a reductive group and its endoscopic group, extending to all blocks and connecting character sheaves and representations.

## Key findings

- Neutral block of Hecke category is equivalent to that of the endoscopic group.
- Relationship between character sheaves with fixed parameters on G and H.
- Correspondence between representations of G and H with fixed semisimple parameters.

## Abstract

For a split 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 $H(\mathbb{F}_{q})$.