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. Results of Leger and Luks and Williams who described H i (n;n) for i 6 2 are generalized: dimH ⁄ (n;⁄ ⁄ (n)) and dimH i (n;n) for i 6 3 are calculated and dimH i (n;n) as function of i and rank g is described for the calssical series.