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October, 2007 $C^{\infty}$-vectors of irreducible representations of exponential solvable Lie groups
J. Math. Soc. Japan 59(4): 1081-1103 (October, 2007). DOI: 10.2969/jmsj/05941081


Let $G$ be an exponential solvable Lie group, and $\pi$ be an irreducible unitary representation of $G$. Then by induction from a unitary character of a connected subgroup, $\pi$ is realized in an $L^2$-space of functions on a homogeneous space. We are concerned with $C^\infty$vectors of $\pi$ from a viewpoint of rapidly decreasing properties. We show that the subspace $\mathscr{PE}$ consisting of vectors with a certain property of rapidly decreasing at infinity can be embedded as the space of the $C^\infty$vectors in an extension of $\pi$ to an exponential group including $G$. Using the space $\mathscr{PE}$, we also give a description of the space $\mathscr{APE}$ related to Fourier transforms of $L^1$-functions on $G$. We next obtain an explicit description of $C^\infty$vectors for a special case. Furthermore, we consider a space of functions on $G$ with a similar rapidly decreasing property and show that it is the space of the $C^\infty$vectors of an irreducible representation of a certain exponential solvable Lie group acting on $L^2(G)$.


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Junko INOUE. Jean LUDWIG. "$C^{\infty}$-vectors of irreducible representations of exponential solvable Lie groups." J. Math. Soc. Japan 59 (4) 1081 - 1103, October, 2007.


Published: October, 2007
First available in Project Euclid: 10 December 2007

zbMATH: 1137.22007
MathSciNet: MR2370007
Digital Object Identifier: 10.2969/jmsj/05941081

Primary: 22E27
Secondary: 22E25 , 43A85

Keywords: $C^{\infty}$-vector , exponential solvable Lie group , unitary representation

Rights: Copyright © 2007 Mathematical Society of Japan

Vol.59 • No. 4 • October, 2007
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