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

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.