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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 HomK(W,V ) possesses the structure of an irreducible U(g) K module whenever HomK(W,V ) is 6= (0). Moreover, dim HomK(W,V ) ≤ 1 for all W and V if and only if U(g) is commutative.