Let $G$ be the adjoint group of a real simple Lie algebra $\mathfrak{ g}_0$ equal either $\mathfrak{ s}\mathfrak{ u}(n,1)$ or $\mathfrak{ s}\mathfrak{ o}(n,1),$ $K$ its maximal compact subgroup, ${ \cal U}(\mathfrak{ g})$ the universal enveloping algebra of the complexification $\mathfrak{ g}$ of $\mathfrak{ g}_0$ and ${ \cal U}(\mathfrak{ g})^K$ its subalgebra of $K-$invariant elements. By a result of F. Knopp [3] ${ \cal U}(\mathfrak{ g})$ is free as a ${ \cal U}(\mathfrak{ g})^K-$module, so there exists a $K-$submodule $E$ of ${ \cal U}(\mathfrak{ g})$ such that the multiplication defines an isomorphism of $K-$modules ${ \cal U}(\mathfrak{ g})^K\otimes E\longrightarrow{ \cal U}(\mathfrak{ g}).$ We prove that $E$ is equivalent to the regular representation of $K,$ i.e. that the multiplicity of every $\delta\in\hat{ K}$ in $E$ equals its dimension. As a consequence we get that for any finitedimensional complex $K-$module $V$ the space $({ \cal U}(\mathfrak{ g})\otimes V)^K$ of $K-$invariants is free ${ \cal U}(\mathfrak{ g})^K-$module of rank $\dim\,V.$
Categories: math.RT