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Monomial Representations of Discrete Type of an Exponential Solvable Lie Group Read chapter Read first chapter

2019 | OriginalPaper | Chapter

An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups

Author : Junko Inoue

Published in: Geometric and Harmonic Analysis on Homogeneous Spaces

Publisher: Springer International Publishing

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Abstract

We discuss a holomorphically induced representation \(\rho =\rho (f,\mathfrak h)\) of Boidol’s group (split oscillator group) G from a real linear form f of the Lie algebra \(\mathfrak g\) of G and a one-dimensional complex subalgebra \(\mathfrak h\) of \(\mathfrak g_\mathbb C\) given by (2.2) in Sect. 2.\(\rho \) is a subrepresentation of the regular representation of G with the Plancherel measure \(\nu \). For \(\nu \)-almost all irreducible representations \(\pi \) of G, the spaces of generalized vectors satisfying the semi-invariance associated with f and \(\mathfrak h\) are one-dimensional subspaces. On the other hand, according to the choice of f, there are two cases that (1) \(\rho \) vanishes, and (2) \(\rho \) is non-zero.

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previous chapter Harmonic Analysis for 4-Dimensional Real Frobenius Lie Algebras
next chapter Spherical Functions for Small K-Types
Literature
1.
Benoist, Y.: Multiplicité un pour les espaces symétriques exponentiels. Mém. Soc. Math. France (N.S.) (15), 1–37 (1984)CrossRef
2.
Bernat, P., Conze, N., Duflo, M., Lévy-Nahas, M., Rais, M., Renouard, P., Vergne, M.: Représentations des groupes de Lie résolubles. Monographies de la Société Mathématique de France, no. 4. Dunod, Paris (1972)
3.
Fujiwara, H.: Représentations monomiales des groupes de Lie résolubles exponentiels. The Orbit Method in Representation Theory, pp. 61–84. Birkhäuser, Basel (1990)CrossRef
4.
Fujiwara, H.: Une réciprocité de Frobenius. Infinite Dimensional Harmonic Analysis III, pp. 17–35. World Scientific Publishing, Hackensack, NJ (2005)CrossRef
5.
Fujiwara, H., Ludwig, J.: Harmonic Analysis on Exponential Solvable Lie Groups. Monographs in Mathematics. Springer, Tokyo (2015)CrossRef
6.
Fujiwara, H., Yamagami, S.: Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel. Representations of Lie groups, Kyoto, Hiroshima, 1986. Advanced Studies in Pure Mathematics, vol. 14, pp. 153–190. Academic Press, Boston, MA (2014)
7.
Inoue, J.: Semi-invariant vectors associated to decompositions of monomial representations of exponential Lie groups. J. Math. Soc. Jpn. 49(4), 647–661 (1997)MathSciNetCrossRef
8.
Inoue, J.: Holomorphically induced representations of some solvable Lie groups. J. Funct. Anal. 186(2), 269–328 (2001)MathSciNetCrossRef
9.
Inoue, J.: Holomorphically induced representations of exponential solvable semi-direct product groups \(\mathbb{R}\ltimes \mathbb{R}^n\). Adv. Pure Appl. Math. 6(2), 113–123 (2015)
10.
Magneron, B.: Représentations induites holomorphes des groupes de Lie nilpotents et involutions complexes. C. R. Acad. Sci. Paris, Ser. I Math. 317, 37–42 (1993)MathSciNetMATH
11.
Magneron, B.: Involutions complexes et vecteurs sphériques associés pour les groupes de Lie nilpotents réels. Astérisque, no. 253 (1999)
12.
13.
14.
Rossi, H., Vergne, M.: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group. J. Funct. Anal. 13, 324–389 (1973)MathSciNetCrossRef
Metadata
Title
An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups
Author
Junko Inoue
Copyright Year
2019
Book
Geometric and Harmonic Analysis on Homogeneous Spaces

Print ISBN: 978-3-030-26561-8

Electronic ISBN: 978-3-030-26562-5

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-26562-5_5

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