A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic

TL;DR

This paper introduces character sheaves and the orbit method for unipotent groups over finite fields, emphasizing their role in understanding the character theory of finite nilpotent groups and groups of the form G(F_q^n).

Contribution

It presents an introduction to the orbit method and character sheaves for unipotent groups, connecting geometric and representation-theoretic perspectives in positive characteristic.

Findings

Introduction of character sheaves for unipotent groups

Explanation of L-packets in the context of finite groups

Insights into character theory for groups G(F_q^n)

Abstract

This article is based on lectures given by the authors in 2005 and 2006. Our first goal is to present an introduction to the orbit method with an emphasis on the character theory of finite nilpotent groups. The second goal (motivated by a recent work of G. Lusztig) is to explain several nontrivial aspects of character theory for finite groups of the form $G(F_{q^n})$, where $G$ is a unipotent algebraic group over a finite field $F_q$. In particular, we introduce the notion of a character sheaf for a unipotent group, and provide a toy model for the representation-theoretic notion of an L-packet.