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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Poisson transforms on vector bundles

Author: An Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 857-887
MSC (1991): Primary 22E46; Secondary 43A85
MathSciNet review: 1370656
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Abstract: Let $G$ be a connected real semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $(\tau,V)$ be an irreducible unitary representation of $K$, and $G\times _K\,V$ the associated vector bundle. In the algebra of invariant differential operators on $G\times _K\,V$ the center of the universal enveloping algebra of $\operatorname{Lie}(G)$ induces a certain commutative subalgebra $Z_\tau$. We are able to determine the characters of $Z_\tau$. Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to $\tau$ the trivial representation.

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Additional Information

An Yang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-251, Cambridge, Massachusetts 02139
Address at time of publication: Micro Strategy, 5th Floor, 2650 Park Tower Dr., Metro Place 1, Vienna, Virginia 22180

Received by editor(s): September 28, 1994
Received by editor(s) in revised form: January 30, 1995
Article copyright: © Copyright 1998 American Mathematical Society