IRREDUCIBLE REPRESENTATIONS OF LIE-ALGEBRAS OF REDUCTIVE GROUPS AND THE KAC-WEISFEILER CONJECTURE

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field of characteristic p > 0. Suppose that G((1)) is simply connected and p is good for the root system of G. If p = 2, suppose in addition that g admits a nondegenerate G-invariant trace form. Let V be an irreducible and faithful g-module with p-character chi is an element of g*. It is proved in the paper that dim V is divisible by p(1/2dim Omega(chi)) where Omega(chi) stands for the orbit of chi under the coadjoint action of G.