ON THE CENTRALIZER OF K IN THE UNIVERSAL ENVELOPING ALGEBRA OF SO(N,1) AND SU(N,1)

Let G(o) be a non compact real semisiple Lie group with finite center, and let U(g)K denote the centralizer in U(g) of a maximal compact subgroup K(o) of G(o). To study the algebra U(g)K, B. Kostant suggested to consider the projection map P: U(g) --> U(t) x U(a), associated to an Iwasawa decomposition G(o) = K(o)A(o)N(o) of G(o), adapted to k(o). When P is restricted to U(g)K J. Lepowsky showed that P becomes an injective anti-homomorphism of U(g)K into U(t)M x U(a). Here U(t)M denotes the centralizer of M(o) in U(t), M(o) being the centralizer of A(o) in K(o). To pursue this idea further it is necessary to have a good characterization of the image of U(g)K in U(t)M x U(a). In this paper we describe such image when G(o) = SO(n, 1)e or SU(n, 1). This is accomplished by establishing a (minimal) set of equations satisfied by the elements in the image of U(g)K, and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations of G(o) given by the Kunze-Stein intertwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case.