A homology vanishing theorem for Kac-Moody algebras with coefficients in the category [formula omitted

Abstract

In this paper, we study the homology of a Kac-Moody Lie-algebra, associated to a symmetrizable generalized Cartan matrix, with coefficients in a module M in the category 0. We show that all the homology groups vanish if the Casimir operator acts as an automorphism on M. In fact we prove a slightly more general theorem (Theorem (1.2)). This is one of the main theorems of this paper. We further study the homology of g with coefficients in arbitrary Verma modules and also integrable highest weight modules. WC prove (Proposition (1.5)) that Hi(g, M(A)) =0 for all i>O and for all the Verma modules M(A) with highest weight ,? E h* such that i is not equal to wp p for any w E W, whereas Hi(g, M( wp p)) w /1 ip ‘(“j(h). This result in the case when g is a finite dimensional semi-simple Lie-algebra is due to Williams. If L(1) is an integrable highest weight module with i #O (2 is automatically dominant integral) then we have, using BGG resolution and Proposition (1.5), that Hi(g, L(;i)) = 0 for all ia 0. This is the content of our Theorem (1.7). In the case when g is finite dimensional, the integrable highest weight modules are precisely the finite dimensional irreducible modules. In this case, this theorem is well known and is due to Whitehead. As in the finite dimensional case, the situation when ,J = 0 (so that L(I1) is the trivial one dimensional module) is drastically different. We have recently proved [Ku; Theorem 1.61 that the homology of the commutator