A TRANSVERSALITY PROPERTY OF A DERIVATION OF THE UNIVERSAL ENVELOPING ALGEBRA U(K), FOR SO(N,1) AND SU(N,1)

Let G0 be a connected real semisimple Lie group and let K0 denote a maximal compact subgroup. Let k subset-of g denote the respective complexified Lie algebras. To study the centralizer U(g)K of K0 in the universal enveloping algebra of g, B.Kostant suggested to consider the projection map P:U(g) --> U(k) x U(a) associated to an Iwasawa decomposition of G0 adapted to K0. To pursue this idea further it is necessary to have a good characterization of the image of U(g)K in U(k) x U(a). If alpha is-an-element-of DELTA+(g,h) is a simple root such that alpha\a not-equal 0, set E(alpha) = X(-alpha) + theta-X(-alpha) is-an-element-of k. Associated to these E(alpha)'s a subalgebra B of U(k) x U(a) was defined and it was proved that P(U(g)K) subset-of B(W approximately). The purpose of this paper is to establish that (SIGMA(j) greater-than-or-equal-to 0 E(alpha)j(U(k)M)) intersect U(k)m+ = 0. This allows to complete the proof of P(U(g)K) = B(W approximately) when G0 = SO(n, 1), SU(n, 1).