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

Let L be an n-dimensional restricted Lie algebra over an algebraically clos ed field K of characteristic p > 0. Given a linear function xi on L and a s calar lambda is an element of K, we introduce an associative algebra U-xi,U - lambda(L) of dimension p(n) over K. The algebra U-xi,U- 0(L) is isomorphi c to the reduced enveloping algebra U-xi(L), while the algebra U-xi,U- (0)( L) is nothing but the reduced symmetric algebra S-xi(L). Deformation argume nts (applied to this family of algebras) enable us to derive a number of re sults on dimensions of simple L-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither suppo rt varieties nor the classification of nilpotent orbits, and compute the ma ximal dimension of simple L-modules for all L having a toral stabiliser of a linear function.