$K-$structure of $\mathcal{U}(\mathfrak{g})$ for $\mathfrak{su}(n,1)$ and $\mathfrak{so}(n,1)$

Abstract

Let $G$ be the adjoint group of a real simple Lie algebra ${\mathfrak{g}}_0$ equal either $\mathfrak{su}(n,1)$ or $\mathfrak{so}(n,1),$ $K$ its maximal compact subgroup, $\mathcal{U}(\mathfrak{g})$ the universal enveloping algebra of the complexification $\mathfrak{g}$ of ${\mathfrak{g}}_0$ and $\mathcal{U}(\mathfrak{g}{)}^K$ its subalgebra of $K-$invariant elements. By a result of F. Knopp [3] $\mathcal{U}(\mathfrak{g})$ is free as a $\mathcal{U}(\mathfrak{g}{)}^K-$module, so there exists a $K-$submodule $E$ of $\mathcal{U}(\mathfrak{g})$ such that the multiplication defines an isomorphism of $K-$modules $\mathcal{U}(\mathfrak{g}{)}^K\otimes E⟶\mathcal{U}(\mathfrak{g}).$ 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 $(\mathcal{U}(\mathfrak{g})\otimes V{)}^K$ of $K-$invariants is free $\mathcal{U}(\mathfrak{g}{)}^K-$module of rank $\dim V.$

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