Lie algebra cohomology and generating functions

Abstract

Let g be a simple Lie algebra, V an irreducible g-module, W the Weyl group and b the Borel subalgebra of g, n = [b, b], h the Cartan subalgebra of g. The Borel-Weil-Bott theorem states that the dimension of H i (n; V ) is equal to the cardinality of the set of elements of length i from W . Here a more detailed description of H i (n; V ) as an h-module is given in terms of generating functions.