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(g)(K)-module whenever Hom(K) (W, V) is not equal (0). Moreover, dimHom(K) (W, V) less than or equal to 1 for all W and V if and only if U(g)(K) is commutative.