Characters of unipotent groups over finite fields

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.

Version 4, 81 pages, LaTeX. Main change compared to the previous version: the term "$L$-packet" has been replaced with "$\mathbb{L}$-packet", which is short for "Lusztig packet" (to distinguish it from Langlands' notion of an $L$-packet)

References in corpus (2)

• A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic
• Base change maps for unipotent algebra groups

Cited by in corpus (1)

• Character sheaves on unipotent groups in positive characteristic: foundations