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The Plancherel formula for the horocycle space and generalizations

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Ronald L. Lipsman
Ronald L. Lipsman
Affiliation:
Department of Mathematics University of MarylandCollege Park, Maryland 20742, USA
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Abstract

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The Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

MSC classification

Secondary: 22E46: Semisimple Lie groups and their representations 43A85: Analysis on homogeneous spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 1 , August 1996 , pp. 42 - 56
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Currey, B., ‘Smooth decomposition of finite multiplicity monomial representations for a class of completely solvable homogeneous spaces’, to appear.Google Scholar
[2]Helgason, S., ‘A duality for symmetric spaces with applications to group representations’, Adv. Math. 5 (1970), 1–154.CrossRefGoogle Scholar
[3]Knapp, A., Representation theory of semisimple Lie groups (Princeton University Press, Princeton, 1986).CrossRefGoogle Scholar
[4]Kostant, B., ‘On the existence and irreducibility of certain series of representations’, in: Lie groups and their representations (Summer school of the Bolyai János Math. Soc.) (Halsted Press, New York, 1975) pp. 231–329.Google Scholar
[5]Lipsman, R., The Penney-Fujiwara Plancherel formula for symmetric spaces, Progress in Math. 82 (Birkhäuser, Boston, 1990) pp. 135–145.Google Scholar
[6]Lipsman, R., ‘The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces’, Pacific J. Math. 151 (1991), 265–295.CrossRefGoogle Scholar
[7]Lipsman, R., ‘The Penney-Fujiwara Plancherel formula for homogeneous spaces’, in: Representation theory of Lie groups and Lie algebras (World Scientific, Singapore, 1992) pp. 120–139.Google Scholar
[8]Lipsman, R., ‘The Plancherel formula for homogeneous spaces with polynomial spectrum’, Pacific J. Math. 159 (1993), 351–377.CrossRefGoogle Scholar
[9]Lipsman, R., ‘The Penny-Fujiwara Plancherel formula for Gelfand paris’, Rocky Mountain J. Math. 26 (1996), 1–23.CrossRefGoogle Scholar
[10]Lipsman, R., ‘Orbital symmetric spaces and finite multiplicity’, J. Funct. Anal. 135 (1996), 1–38.CrossRefGoogle Scholar