Representations of restricted Lie algebras and families of associative ℒ-algebras

Abstract

Let ℒ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K, we introduce an associative algebra Uξ,λ(ℒ) of dimension pn over K. The algebra Uξ,1(ℒ) is isomorphic to the reduced enveloping algebra Uξ(ℒ), while the algebra Uξ,0(ℒ) is nothing but the reduced symmetric algebra Sξ(ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function.