Kostant's formula for a certain class of generalized Kac-Moody algebras

Abstract

Let \mathfrak{g}(A) be a symmetrizable generalized Kac-Moody algebra with \mathfrak{h} its Cartan subalgebra and \mathfrak{n}\underbar{\phantom{a}} the sum of all its negative root spaces. In this paper, we prove the generalization of Kostant's homology formula under a certain condition on the matrix A. This formula completely determines the \mathfrak{h}-modules structure of the homology of \mathfrak{n}\underbar{\phantom{a}} in the irreducible highest weight \mathfrak{g}(A)-module L(Λ) with an arbitrary dominant integral weight Λ.