A proof of the first Kac-Weisfeiler conjecture in large characteristics

TL;DR

This paper proves the first Kac-Weisfeiler conjecture for restricted Lie algebras in large characteristic fields, confirming the maximal dimension of simple modules in these cases using classical Lie theory techniques.

Contribution

It provides the first proof of the conjecture for all restricted Lie subalgebras of isplaystyle gl_n(k) in large characteristic fields, extending to Lie algebras of group schemes.

Findings

The conjecture holds for restricted Lie subalgebras of isplaystyle gl_n(k) in large characteristic.

The result applies to Lie algebras of group schemes over algebraically closed fields.

A short proof is provided for Lie algebras over finitely generated rings in characteristic zero.

Abstract

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p \gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring $R \subseteq \mathbb{C}$, after base change to a field of large positive characteristic.