Restrictions from gln to sln

Let K be an algebraically closed field, let n be a positive integer.Consider the general linear Lie algebra of all (n n) -matrices over K and its subalgebra of all matrices with trace equal to 0 , the special linear Lie algebra.If the characteristic of K does not divide n , then the larger Lie algebra is the direct product of the smaller Lie algebra with a one dimensional Lie algebra; in this case each finite dimensional simple module for the general linear Lie algebra restricts to a simple module for the special linear Lie algebra.This is no longer the case when the characteristic of K divides n ; the purpose of this paper is to describe what happens in this situation.