A strong generalization of Helgason’s theorem

Abstract:

Let $G$ be a simple Lie group with $KAN$ an Iwasawa decomposition of $G$, and let $M$ be the centralizer of $A$ in $K$. Suppose ${K_1}$ is a fixed, closed, normal, analytic subgroup of $K$, and set ${\mathbf {P}}({K_1})$ equal to the set of all parabolic subgroups $P$ of $G$ which contain $MAN$ such that ${K_1}P = G$ and ${K_1} \cap P$ is normal in the reductive part of $P$. Suppose $\pi :G \to GL(V)$ is an irreducible representation of $G$. Then, if ${\mathbf {P}}({K_1}) \ne \emptyset$, we obtain necessary and sufficient conditions for ${V^{{K_1}}}$, the space of ${K_1}$-fixed vectors, to be $\ne (0)$. Moreover, reciprocity formulas are obtained which determine $\dim {V^{{K_1}}}$.