Kac–Weisfeiler's maximal simple-module dimension conjecture

Let $\mathfrak{g}$ be a restricted Lie algebra, and let $M(\mathfrak{g})$ denote the maximal dimension of a simple $U(\mathfrak{g})$-module.

Kac–Weisfeiler conjecture (KW1).

$$M(\mathfrak{g})=p^{\frac{1}{2}(\dim(\mathfrak{g})-\operatorname{ind}(\mathfrak{g}))}.$$

This is a modular representation-theoretic conjecture concerning the largest dimension of a simple module over the restricted enveloping algebra. It remains open in general, although it is known for several classes of restricted Lie algebras, and the paper proves it for the truncated current Lie algebras under consideration in the stated reductive-group setting.