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Irreducible representations of the group of movements of the Euclidean plane

Published online by Cambridge University Press:  09 April 2009

James Symons
James Symons
Affiliation:
Monash University, Clayton, Vic. 3168
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It is well known that a wide range of Special Function Theory can be realized by considering unitary representations of certain topological groups.

In this approach it is very important to determine all irreducible continuous unitary representations of the group in question.

For the group of movements this problem was initiated by Vilenkin [6]. Rather restrictive conditions were imposed in this paper and while he returned to the problem in [7], it was still not solved in full generality (among other things the representation space was assumed separable). The first complete solution appears to have been given by Thoma [5]. Here, the method was to show each irreducible continuous unitary representation equivalent to a particular representation in a space of square integrable functions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 1 , August 1974 , pp. 78 - 96
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Bingen, F., ‘Les fonctions de Bessel d'ordre entire et les représentations du group des placements du plan’, Bull. Soc. Math. de Belg. (1965), 115–152.Google Scholar
[2]Gelfand, and Shapiro, , Representations of the group of rotations (Reprinted in A. M. S. Translations Series 2, Vol. 2, 1956).Google Scholar
[3]Hille, and Phillips, , Functional Analysis and Semigroups (A. M. S. 1957).Google Scholar
[4]Naimark, M. A., Normed Rings (Noordhoff 1964).Google Scholar
[5]Thoma, E., ‘Die unitaren Darsettellungen der unwersellen Überlagerungsgruppe der Bewegungsgruppe des R2’, Math. Ann. 134 (1958), 428–459.CrossRefGoogle Scholar
[6]Vilenkin, N. J., Bessel functions and representations of the group of Euclidean movements (U. M. N. 1956 (69) Russian).Google Scholar
[7]Vilenkin, N. J., Special Functions and the Theory of Group Representations (A. M. S. 1968).CrossRefGoogle Scholar
[8]Akhiezer, and Glazman, . Theory of linear operators in Hilbert Space. Volume 1. (Ungar 1966).Google Scholar