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 $γ_1 <\cdots<γ_r$ be the Harish-Chandra strongly orthogonal roots, with $γ_1$ the lowest non-compact root, and let $ζ$ be the normalized central character of $\mathfrak k$. We consider the unitary highest we