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

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.

77 pages, LaTeX. A few proofs are missing from this version of the paper (only the essential ideas are sketched), but otherwise it is complete. Main change compared with the first version: the term "$L$-packet" has been replaced with "$\mathbb{L}$-packet" (which is short for "Lusztig packet")

References in corpus (2)

• Irreducible Characters of Finite Algebra Groups
• Base change maps for unipotent algebra groups

Cited by in corpus (21)

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• Superinduction for pattern groups
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• Metric groups attached to biextensions
• Motivic Poisson summation
• Heisenberg characters, unitriangular groups, and Fibonacci numbers
• Modular Categories Associated to Unipotent Groups
• Restricting supercharacters of the finite group of unipotent uppertriangular matrices
• Characters of unipotent groups over finite fields
• Hasse-Arf filtrations in $p$-adic analytic geometry -- Fourth Release
• Branching rules in the ring of superclass functions of unipotent upper-triangular matrices
• Minimal Idempotents on Solvable Groups
• Orthogonal subsets of root systems and the orbit method
• Compatible intertwiners for representations of finite nilpotent groups
• The orbit method for profinite groups and a p-adic analogue of Brown's theorem
• Exponentiation of commuting nilpotent varieties
• Stacky Abelianization of an Algebraic Group
• Modular categories, orbit method and character sheaves on unipotent groups
• Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution