Journal of Lie Theory 21 (2011), No. 1, 165--188
Copyright Heldermann Verlag 2011
The Image of the Lepowsky Homomorphism for SO(n,1) and SU(n,1)
Alfredo Brega
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
brega@mate.uncor.edu
Leandro Cagliero
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
cagliero@mate.uncor.edu
Juan Tirao
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
tirao@mate.uncor.edu
[Abstract-pdf]
\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$(n,1)$.
Keywords: Semisimple Lie groups, universal enveloping algebra, representation theory, group invariants, restriction theorem, Kostant degree.
MSC: 22E46, 16S30, 16U70
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