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

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 $χ$ on $\mathfrak{g}$ we denote by $U_χ(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $χ$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_χ(\mathfrak{g})$-module has dimension divisible by $p^{d(χ)}$ where $2d(χ)$ is the dimension of the coadjoint $G$-orbit containing $χ$. 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 $U_χ(\mathfrak{g})$ admits a module of dimension $p^{d(χ)}$.