Modular representations of Lie algebras of reductive groups and Humphreys' conjecture

TL;DR

This paper proves that for certain reductive Lie algebras over fields of positive characteristic, the reduced enveloping algebra admits modules of minimal possible dimension, confirming a longstanding conjecture.

Contribution

It provides a positive answer to Humphreys' conjecture, showing the existence of modules with minimal dimension for all relevant reduced enveloping algebras.

Findings

Existence of modules with dimension p^{d(χ)} for all reduced enveloping algebras.

Confirmation of Humphreys' conjecture from the 1990s.

Advancement in understanding modular representations of Lie algebras.

Abstract

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $\chi$ on $\mathfrak{g}$ we denote by $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_\chi(\mathfrak{g})$-module has dimension divisible by $p^{d(\chi)}$ where $2d(\chi)$ is the dimension of the coadjoint $G$-orbit containing $\chi$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_\chi(\mathfrak{g})$ admits a module of dimension $p^{d(\chi)}$.