A note on branching theorems

Let $G$ be a complex, simply connected semisimple analytic group with $K$ a closed connected reductive subgroup. Suppose $V$ is an irreducible holomorphic $G$-module and $W$ an irreducible holomorphic $K$-module. We prove that Hom$_{K}(W,V)$ possesses the structure of an irreducible $U(\mathfrak{g})^{K}$-module whenever $\text{Hom}_{K}(W,V)$ is $\neq (0)$. Moreover, $\dim\text{Hom}_{K} (W,V)\le 1$ for all $W$ and $V$ if and only if $U{(\mathfrak{g})}^{K}$ is commutative.