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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

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