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Transactions of the American Mathematical Society

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



A strong generalization of Helgason's theorem

Author: Kenneth D. Johnson
Journal: Trans. Amer. Math. Soc. 304 (1987), 171-192
MSC: Primary 22E46
MathSciNet review: 906811
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Abstract: Let $ G$ be a simple Lie group with $ KAN$ an Iwasawa decomposition of $ G$, and let $ M$ be the centralizer of $ A$ in $ K$. Suppose $ {K_1}$ is a fixed, closed, normal, analytic subgroup of $ K$, and set $ {\mathbf{P}}({K_1})$ equal to the set of all parabolic subgroups $ P$ of $ G$ which contain $ MAN$ such that $ {K_1}P = G$ and $ {K_1} \cap P$ is normal in the reductive part of $ P$. Suppose $ \pi :G \to GL(V)$ is an irreducible representation of $ G$. Then, if $ {\mathbf{P}}({K_1}) \ne \emptyset $, we obtain necessary and sufficient conditions for $ {V^{{K_1}}}$, the space of $ {K_1}$-fixed vectors, to be $ \ne (0)$. Moreover, reciprocity formulas are obtained which determine $ \dim {V^{{K_1}}}$.

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Keywords: Reciprocity pair, parabolic group, representation
Article copyright: © Copyright 1987 American Mathematical Society

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