On the $K$-types of certain families of unitary highest weight representations of Hermitian Lie groups

Abstract

Let $(\mathfrak g, \mathfrak k)$ be the complexication of a Hermitian symmetric pair and let $\gamma_1<\cdots<\gamma_r$ be the Harish-Chandra strongly orthogonal roots, with $\gamma_1$ the lowest non-compact root, and let $\zeta$ be the normalized central character of $\mathfrak k$. We consider the unitary highest weight representations $L(\lambda(x))$ of $\mathfrak g $ of highest weights of the form $\lambda(x)=-k\gamma_1 -x\zeta$, $x\in\mathbb{R}$. In particular, we are interested in the case $x=a$, the last reduction point in the analytic continuation in $x$. We prove that $L(-k\gamma_1-a \zeta)$, $k\ge 1$, is $K$-multiplicity free and we describe all its K-types.