The image of the Lepowsky homomorphism for the split rank one symplectic group

Let Go be a semisimple Lie group and let Ko denote a maximal compact subgroup of Go. Let U(g) be the complex universal enveloping algebra of Go and let UK(g) denote the centralizer of Ko in U(g). Also let be the projection map corresponding to the direct sum U(g)=(U(k)⊗U(a))⊕U(g)n associated to an Iwasawa decomposition of Go adapted to Ko. In this paper we give a characterization of the image of UK(g) under the injective antihomomorphism , considered by Lepowsky in [J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973) 1–44], when Go=Sp(n,1).