A Positive Characteristic Resolution Problem for SL(3, k)

Abstract

In 1975,1resolved 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 BGG) resolution. One consequence of the BGG 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 BGG 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 BGG proof of the Weyl character formula.