The image of the Lepowsky homomorphism for SO(n,1) and SU(n,1)

OSCAR BREGA; LEANDRO CAGLIERO; JUAN TIRAO

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Año: 2011 vol. 21 p. 165 - 188

0949-5932

Let $G_o$ be a classical rank one semisimple Lie group and let $K_o$ denote a maximal compact subgroup of $G_o$. Let $U(lieg)$ be the complex universal enveloping algebra of $G_o$ and let $U(lieg)^K$ denote the centralizer of $K_o$ in $U(lieg)$. Also let $P:U(lieg) longrightarrow U(liek) otimes U(liea)$ be the projection map corresponding to the direct sum $U(lieg)= (U(liek) otimes U(liea)) oplus U(lieg) lien$ associated to an Iwasawa decomposition of $G_o$ adapted to $K_o$. In this paper we give a characterization of the image of $U(lieg)^K$ under the injective antihomorphism $P:U(lieg)^K longrightarrow U(liek)^M otimes U(liea)$ when $G_o$ is locally isomorphic to SO$(n,1)$ and SU$(n,1)$.