The Image of the Lepowsky Homomorphism for SO(n,1) and SU(n,1)..

ALFREDO BREGA; LEANDRO CAGLIERO; JUAN TIRAO

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Año: 2011 vol. 21 p. 165 - 188

0949-5932

Let G0 be a semisimple Lie group, let K0 be a maximal compact subgroup of G0, and let k and g denote the complexification of  their Lie algebras. Let G be the adjoint group of  g and let K be the connected Lie subgroup of G with Lie algebra ad(k). If U(g) is the universal enveloping algebra of g then U(g)K will denote the centralizer of K in U(g). Also let P: U(g) → U(k) x U(a) be the projection map corresponding to the direct sum U(g) = ( U(k) x U(a)) + U(g)n associated to an Iwasawa decomposition of G0 adapted to K0. In this paper we give a characterization of the image of U(g)K under the injective antihomorphism P: U(g)K → U(k)M x U(a) when G0  is locally isomorphic to SO(n,1) and SU(n,1).