On the dimensions of the irreducible modules of Lie algebras of classical type

Abstract

Introduction. In this and another paper [4], partial answers are given to two of the questions raised in ?11.5 of [3]. Because the results in [4] hold for a more extensive class of Lie algebras of classical type than those considered in this paper, the two questions have been treated separately. Let V be a Lie algebra over an algebraically closed field Q of characteristic p > 7 whose Killing form is nondegenerate, and let M be an irreducible restricted right V-module. We shall give a computable sufficient condition on M in order for Weyl's formula [9, p. 359], suitably interpreted, to give the dimension of M over Q. The problem of calculating the dimension of M in all cases remains unsolved. In the last section we obtain an upper bound for the dimension of any irreducible restricted right V-module, and from this result it follows that Weyl's formula as we have interpreted it does not always give the dimension of M. 1. Definitions and preliminary results. Familiarity with the papers [1; 2] and [3] is assumed. First we list some notations.