The structure of the algebra $(\cU(\gg)\otim C(\pp))^K$ for the groups $\SU(n,1)$ and $\SO_e(n,1)$

Abstract

The structure of the algebra of $K-$invariants in $\cU(\gg)\otim C(\pp)$ is important for constructing $(\gg,K)-$modules by means of algebraic Dirac induction as developed in [5] and its variants in [8] and [10]. We show that for the groups $\SU(n,1)$ and $\SO_e(n,1)$ this algebra is a free $\cU(\gg)^K-$module of rank $\dim C(\pp)=2^{\dim\pp}.$ We also indicate a way of constructing a $\cU(\gg)^K-$basis in $(\cU(\gg)\otim C(\pp))^K.$