Characters of unipotent groups over finite fields

TL;DR

This paper defines L-packets for irreducible representations of unipotent groups over finite fields, describes them explicitly, and confirms a conjecture that their dimensions are powers of q under certain conditions.

Contribution

It introduces a new definition of L-packets for unipotent groups over finite fields and provides an explicit description in terms of admissible pairs, linking representation theory and geometry.

Findings

L-packets are explicitly described via admissible pairs.

Confirmed that irreducible representation dimensions are powers of q under connected centralizer condition.

First in a series exploring representations, geometry, and character sheaves of unipotent groups.

Abstract

Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called "admissible pairs" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F_q) (where G is a unipotent algebraic group over F_q), the geometry of G, and the theory of character sheaves.