Journal of the Mathematical Society of Japan

$C^{\infty}$-vectors of irreducible representations of exponential solvable Lie groups

Junko INOUE and Jean LUDWIG

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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 $L^2(G)$.

Article information

J. Math. Soc. Japan, Volume 59, Number 4 (2007), 1081-1103.

First available in Project Euclid: 10 December 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Secondary: 22E25: Nilpotent and solvable Lie groups 43A85: Analysis on homogeneous spaces

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


INOUE, Junko; LUDWIG, Jean. $C^{\infty}$-vectors of irreducible representations of exponential solvable Lie groups. J. Math. Soc. Japan 59 (2007), no. 4, 1081--1103. doi:10.2969/jmsj/05941081.

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