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 be the complex universal enveloping algebra of Go and let denote the centralizer of Ko in . Also let be the projection map corresponding to the direct sum associated to an Iwasawa decomposition of Go adapted to Ko. In this paper we give a characterization of the image of 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).