Poisson transforms on vector bundles

Author:
An Yang

Journal:
Trans. Amer. Math. Soc. **350** (1998), 857-887

MSC (1991):
Primary 22E46; Secondary 43A85

DOI:
https://doi.org/10.1090/S0002-9947-98-01659-6

MathSciNet review:
1370656

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Abstract: Let be a connected real semisimple Lie group with finite center, and a maximal compact subgroup of . Let be an irreducible unitary representation of , and the associated vector bundle. In the algebra of invariant differential operators on the center of the universal enveloping algebra of induces a certain commutative subalgebra . We are able to determine the characters of . 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 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

Email:
yang@strategy.com

DOI:
https://doi.org/10.1090/S0002-9947-98-01659-6

Received by editor(s):
September 28, 1994

Received by editor(s) in revised form:
January 30, 1995

Article copyright:
© Copyright 1998
American Mathematical Society