Laplace operators and the $\mathfrak {h}$ module structure of certain cohomology groups

Abstract:

Let $\mathfrak {n}$ be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra $\mathfrak {g}$. Under the adjoint action $\mathfrak {n},\mathfrak {g}/\mathfrak {n}$, and $\mathfrak {n}’$ (the dual space of $\mathfrak {n}$) are $\mathfrak {n}$ modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of $\mathfrak {n}$ with coefficients in $\mathfrak {n},\mathfrak {g}/\mathfrak {n}$, and $\mathfrak {n}’$ is obtained. These cohomology groups (spaces) are described, in fact, as completely reducible modules of a Cartan subalgebra $\mathfrak {h}$ of $\mathfrak {g}$.