Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-01T22:06:20.027Z Has data issue: false hasContentIssue false
  • English
  • Français

Square-integrable representations of non-unimodular groups

Published online by Cambridge University Press:  17 April 2009

A.L. Carey
A.L. Carey
Affiliation:
Department of Mathematical Physics, University of Adelaide, Adelaide, South Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 1 , August 1976 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Carey, A.L., “Group representations in reproducing kernel Hilbert spaces” (Preprint, Department of Mathematical Physics, University of Adelaide, Adelaide, 1975).Google Scholar
[2]Dixmier, Jacques, Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1964).Google Scholar
[3]Duflo, M. and Moore, Calvin C., “On the regular representation of a nonunimodular locally compact group”, J. Functional Analysis 21 (1976), 209–243.Google Scholar
[4]Gaal, Steven A., Linear analysis and representation theory (Die Grundlehren der mathematischen Wissenschaften, 198. Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[5]Godement, Roger, “Sur les relations d'orthogonalité de V. Bargmann. I. Résultats préliminaires”, C.R. Acad. Sci. 225 (1947), 521–523.Google Scholar
[6]Godement, Roger, “Sur les relations d'orthogonalité de V. Bargmann. II. Démonstration générale”, C.R. Acad. Sci. 225 (1947), 657–659.Google Scholar
[7]Godement, Roger, “Les fonctions de type positif et la théorie des groupes”, Trans. Amer. Math. Soc. 63 (1948), 1–84.Google Scholar
[8]Kleppner, Adam and Lipsman, Ronald L., “The Plancherel formula for group extensions II”, Ann. Sci. École. Norm. Sup. (4) 6 (1973), 103–132.Google Scholar
[9]Kreĭn, M.G., “Hermitian-positive kernels on homogeneous spaces. I”, Amer. Math. Soc. Transl. (2) 34 (1963), 69–108.Google Scholar
[10]Kreĭn, M.G., “Hermitian-positive kernels on homogeneous spaces. II”, Amer. Math. Soc. Transl. (2) 34 (1963), 109–164.Google Scholar
[11]Phillips, John, “A note on square-integrable representations”, J. Functional Analysis 20 (1975), 83–92.Google Scholar
[12]Segal, I.E., “An extension of Plancherel's formula to separable unimodular groups”, Ann. of Math. (2) 52 (1950), 272–292.Google Scholar