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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Abstract

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

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

Dates
First available in Project Euclid: 10 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1197320629

Digital Object Identifier
doi:10.2969/jmsj/05941081

Mathematical Reviews number (MathSciNet)
MR2370007

Zentralblatt MATH identifier
1137.22007

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.jmsj/1197320629


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References

  • [1] J. Andele, Noyaux d'opérateurs sur les groupes de Lie exponentiels, Thèse, Université de Metz, 1997.
  • [2] \auD. Arnal, H. Fujiwara and J. Ludwig, Opérateurs d'entrelacement pour les groupes de Lie exponentiels, \tiAmer. J. Math., , 118 ((1996),)\spg839–\epg878.
  • [3] P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Rais, P. Renouard and M. Vergne, Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France, Dunod, Paris, 1972.
  • [4] \auL. Corwin, F. P. Greenleaf and R. Penney, A general character formula for irreducible projections on $L^{2}$ of a nilmanifold, \tiMath. Ann., , 225 ((1977),)\spg21–\epg32.
  • [5] \auA. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk, 17 (1962), 57–110, \titranslated to: Russ. Math. Surv., , 17 ((1962),)\spg53–\epg104.
  • [6] H. Leptin and J. Ludwig, Unitary representation theory of exponential Lie groups, de Gruyter, Berlin, 1994.
  • [7] \auJ. Ludwig, Irreducible representations of exponential solvable Lie groups and operators with smooth kernels, \tiJ. Reine Angew. Math., , 339 ((1983),)\spg1–\epg26.
  • [8] \auN. S. Poulsen, On $C^{\infty}$-vectors and intertwining bilinear forms for representations of Lie groups, \tiJ. Functional Analysis, , 9 ((1972),)\spg87–\epg120.
  • [9] L. Pukanszky, On a property of the quantization map for the coadjoint orbits of connected Lie groups, The orbit method in representation theory, Copenhagen, 1988, Progr. Math., 82, Birkhäuser Boston, Boston, MA, 1990, pp. 187–211.