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

Mitya Boyarchenko, Vladimir Drinfeld|arXiv (Cornell University)|Sep 27, 2006

Finite Group Theory Research26 references45 citations

TL;DR

This paper introduces character sheaves and ${\mathbb{L}}$-packets for unipotent groups over finite fields in positive characteristic, extending the orbit method beyond characteristic zero. It establishes that character sheaves arise as Fourier transforms of equivariant local systems on coadjoint orbits, even when the orbit method fails, and identifies new phenomena such as disconnected stabilizers and odd-dimensional orbits.

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.

Motivation & Objective

• To develop a geometric framework for character theory of finite unipotent groups $G(\mathbb{F}_{q^n})$ over finite fields.
• To generalize the orbit method to positive characteristic, particularly for unipotent groups with nilpotence class $\geq p$.
• To define character sheaves and ${\mathbb{L}}$-packets independently of the orbit method, enabling applications beyond the classical setting.
• To clarify key differences from characteristic zero representation theory, such as non-trivial stabilizers and non-integer-dimensional orbits.
• To provide a foundation for geometric representation theory of unipotent groups via sheaf-theoretic methods.

Proposed method

• Define character sheaves on $G \otimes_{\mathbb{F}_q} \overline{\mathbb{F}}_q$ as inverse Fourier transforms of irreducible equivariant local systems on coadjoint orbits.
• Use the Lie ring scheme $\mathfrak{g}$ associated to a unipotent group $G$ over $\mathbb{F}_q$, even when $G$ is not isomorphic to its Lie algebra.
• Introduce ${\mathbb{L}}$-packets as collections of irreducible representations associated to a single geometric orbit, especially when stabilizers are disconnected.
• Apply the Campbell-Hausdorff formula in positive characteristic to analyze logarithmic and exponential maps on Lie rings.
• Use equivariant derived categories and Fourier-Deligne transforms to relate sheaf-theoretic constructions to representation theory.
• Construct counterexamples to show that standard orbit method properties fail in positive characteristic, such as non-linear combinations of characters in orbit functions.

Experimental results

Research questions

• RQ1 How can the orbit method be extended to unipotent groups over finite fields in positive characteristic, especially when the nilpotence class is $\geq p$?
• RQ2 What is the correct geometric definition of a character sheaf for unipotent groups in positive characteristic, independent of the orbit method?
• RQ3 Why do coadjoint orbits in positive characteristic sometimes have odd dimension, and how does this affect representation theory?
• RQ4 Under what conditions does a geometric orbit correspond to multiple irreducible representations (i.e., an ${\mathbb{L}}$-packet)?
• RQ5 Can the theory of character sheaves be used to understand the representation theory of $UL_{N,q}$, the unipotent upper-triangular matrices, when $N \gg p$?

Key findings

• Character sheaves on $G \otimes_{\mathbb{F}_q} \overline{\mathbb{F}}_q$ are (up to cohomological shift) the inverse Fourier transforms of irreducible equivariant local systems on coadjoint orbits, even when the orbit method does not apply.
• The stabilizer of a point in $\mathfrak{g}^*$ may be disconnected, leading to geometric orbits that correspond to multiple irreducible representations—forming an ${\mathbb{L}}$-packet.
• Coadjoint orbits in positive characteristic can have odd dimension, a phenomenon absent in characteristic zero.
• The function $x \mapsto \lambda(\log(\gamma e^x))$ is not always a linear combination of characters from the $\Gamma$-orbit, violating a key property of the classical orbit method.
• Counterexamples show that the Campbell-Hausdorff formula in positive characteristic leads to non-linear terms (e.g., $c_2 = 1/12$) that prevent standard character decompositions.
• Du Cloux's theorem in characteristic zero is generalized: $S(\mathfrak{g})/J(\Omega) \cong U(\mathfrak{g})/I(\Omega)$ as $\mathfrak{g}$-modules iff $\dim \Omega = 2$ and $\Omega$ has degree $\leq 2$.