Poisson transforms on vector bundles

Yang, An

doi:10.1090/S0002-9947-98-01659-6

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 $G$ 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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