Character Sheaves of Algebraic Groups Defined over Non-Archimedean Local Fields

TL;DR

This paper explores the relationship between character sheaves of algebraic groups over non-Archimedean local fields and the characters of their smooth representations, providing new insights and examples for general linear groups.

Contribution

It extends the theory of character sheaves to non-Archimedean local fields and demonstrates how to match these sheaves with virtual representations, including explicit examples.

Findings

Established a correspondence between character sheaves and virtual representations over non-Archimedean fields

Provided explicit examples for general linear groups demonstrating the matching process

Enhanced understanding of character sheaves in the context of p-adic groups

Abstract

This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations.