\def\a{{\frak a}} \def\g{{\frak g}} \def\k{{\frak k}} \def\n{{\frak n}} Let $G_o$ be a classical rank one semisimple Lie group and let $K_o$ denote a maximal compact subgroup of $G_o$. Let $U(\g)$ be the complex universal enveloping algebra of $G_o$ and let $U (\g)^K$ denote the centralizer of $K_o$ in $U(\g)$. Also let $P:U(\g)\longrightarrow U(\k)\otimes U(\a)$ be the projection map corresponding to the direct sum $U(\g) = \bigl(U(\k)\otimes U(\a)\bigr)\oplus U(\g)\n$ associated to an Iwasawa decomposition of $G_o$ adapted to $K_o$. In this paper we give a characterization of the image of $U(\g)^K$ under the injective antihomorphism $P:U(\g)^K\longrightarrow U(\k)^M\otimes U(\a)$ when $G_o$ is locally isomorphic to SO$(n,1)$ and SU$(n,1)$.
1
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
Alfredo Brega; Leandro Cagliero; Juan Tirao. The Image of the Lepowsky Homomorphism for SO(n,1) and SU(n,1). Journal of Lie Theory, Tome 21 (2011) no. 1, pp. 165-188. http://geodesic.mathdoc.fr/item/JOLT_2011_21_1_a7/
@article{JOLT_2011_21_1_a7,
author = {Alfredo Brega and Leandro Cagliero and Juan Tirao},
title = {The {Image} of the {Lepowsky} {Homomorphism} for {SO(n,1)} and {SU(n,1)}},
journal = {Journal of Lie Theory},
pages = {165--188},
year = {2011},
volume = {21},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JOLT_2011_21_1_a7/}
}
TY - JOUR
AU - Alfredo Brega
AU - Leandro Cagliero
AU - Juan Tirao
TI - The Image of the Lepowsky Homomorphism for SO(n,1) and SU(n,1)
JO - Journal of Lie Theory
PY - 2011
SP - 165
EP - 188
VL - 21
IS - 1
UR - http://geodesic.mathdoc.fr/item/JOLT_2011_21_1_a7/
ID - JOLT_2011_21_1_a7
ER -
%0 Journal Article
%A Alfredo Brega
%A Leandro Cagliero
%A Juan Tirao
%T The Image of the Lepowsky Homomorphism for SO(n,1) and SU(n,1)
%J Journal of Lie Theory
%D 2011
%P 165-188
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/JOLT_2011_21_1_a7/
%F JOLT_2011_21_1_a7
