A positive characteristic resolution problem for SL(3, k)

In 1975, I. N. Bernstein, I. M. Gel'fand, and S. I. Gel'fand (in "Lie Groups and Their Representations," pp. 21-64, Halsted, New York, 1975) resolved an irreducible representation of a complex semisimple Lie algebra by Verma modules indexed by the Weyl group. This resolution is now commonly referred to as the Bernstein-Gel'fand-Gel'fand (or EGG) resolution. One consequence of the EGG resolution is a simple proof of the Weyl character formula. In this paper, we will describe an analogous resolution problem in positive characteristic: Is there a resolution of a highest weight irreducible representation (of a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic) by restricted Verma modules? And if so, is it a generalization of the EGG resolution? This paper provides a complete answer to this problem for SL(3, k). Consequently, we are able to compute the formal character of the irreducible representation following a procedure similar to the EGG proof of the Weyl character formula. (C) 1997 Academic Press.