Kostant's formula for Kac-Moody Lie algebras

Abstract

This paper is a generalization of H. Garland and J. Lepowsky's paper of 1976. Let g( A) be the Kac-Moody Lie algebra defined by a symmetrizable generalized Cartan matrix A, M(λ) the irreducible highest weight module of g( A) with a dominant integral highest weight λ Kostant's homology and cohomology formulas for the Lie algebra g( A) and the module M(λ) are proved without assuming that the reductive part of the parabolic subalgebra of g( A) is of finite type. A resolution of Mλ is constructed. For any j, the jth term of the resolution has a filtration such that all the factors of the filtration are generalized Verma modules of the form V m( w( λ + p)− p) where w ranges over a certain subset of the Weyl group of g( A).