Lie algebra cohomology and generating functions

Abstract

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

Results of Leger and Luks and Williams who described $H^{i}(\mathfrak{n}; \mathfrak{n})$ for $i\leq 2$ are generalized: $\dim H^{*}(\mathfrak{n}; \Lambda^{*}(\mathfrak{n}))$ and $\dim H^{i}(\mathfrak{n}; \mathfrak{n})$ for $i\leq 3$ are calculated and $\dim H^{i}(\mathfrak{n}; \mathfrak{n})$ as function of $i$ and rank $\mathfrak{g}$ is described for the calssical series.