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

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 sufficiently large characteristic $p$ (depending on $n$). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic.

In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of a group scheme over a finitely generated ring $R \subseteq \mathbb {C}$, after base change to a field of large positive characteristic.