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Transactions of the American Mathematical Society

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



Ultra-irreducibility of induced representations of semidirect products

Author: Henrik Stetkær
Journal: Trans. Amer. Math. Soc. 324 (1991), 543-554
MSC: Primary 22E45; Secondary 22D30
MathSciNet review: 974525
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Abstract: Let the Lie group $G$ be a semidirect product, $G = SK$, of a connected, closed, normal subgroup $S$ and a closed subgroup $K$. Let $\Lambda$ be a nonunitary character of $S$, and let ${K_\Lambda }$ be its stability subgroup in $K$. Let ${I^{\Lambda \mu }}$, for any irreducible representation $\mu$ of ${K_\Lambda }$, denote the representation ${I^{\Lambda \mu }}$ of $G$ induced by the representation $\Lambda \mu$ of $S{K_\Lambda }$. The representation spaces are subspaces of the distributions. We show that ${I^{\Lambda \mu }}$ is ultra-irreducible when the corresponding Poisson transform is injective, and find a sufficient condition for this injectivity.

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Keywords: Lie group, semidirect product, nonunitary representation, induced representation, ultra-irreducibility, Poisson transform, distribution space
Article copyright: © Copyright 1991 American Mathematical Society