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Analyse conforme sur les algèbres de Jordan

Part of: Harmonic analysis in several variables Jordan algebras

Published online by Cambridge University Press:  09 April 2009

M. Pevzner
M. Pevzner
Affiliation:
Université Libre de Bruxelles, CP 218, Campus de la Plaine, 1050 Brussels, Belgium e-mail: [email protected]
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Abstract

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We construct the Weil representation of the Kantor-Koecher-Tits Lie algebra g associated to a simple real Jordan algebra V. Later we introduce a family of integral operators intertwining the Weil representation with the infinitesimal representations of the degenerate principal series of the conformal group G of the Jordan algebra V. The decomposition of L2(V) in the case of Jordan algebra of real square matrices is given using this construction.

MSC classification

Secondary: 42B35: Function spaces arising in harmonic analysis 17C30: Associated groups, automorphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 2 , October 2002 , pp. 279 - 300
Copyright
Copyright © Australian Mathematical Society 2002

References

Références bibliographiques

[1]Bertram, W., ‘On some causal and conformal groups’, J. Lie Theory (2) 6 (1996), 215–247.Google Scholar
[2]Bertram, W., ‘Un théorème de Liouville pour les algèbres de Jordan’, Bull. Soc. Math. France (2) 124 (1996), 299–327.CrossRefGoogle Scholar
[3]Clerc, J.-L., ‘A generalized Hecke identity’, J. Fourier Anal. Appl. (1) 6 (2000), 105–111.CrossRefGoogle Scholar
[4]Faraut, J. and Gindikin, S., ‘Pseudo-Hermitian symmetric spaces of tube type’, in: Topics in geometry (Birkhäuser Boston, Boston, MA, 1996) pp. 123–154.CrossRefGoogle Scholar
[5]Faraut, J. and Korányi, A., Analysis on symmetric cones, Oxford Math. Monographs. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1994).CrossRefGoogle Scholar
[6]Herz, C. S., ‘Bessel functions of matrix argument’, Ann. of Math. (2) 61 (1955), 474–523.CrossRefGoogle Scholar
[7]Johnson, K. D., ‘Degenerate principal series and compact groups’, Math. Ann. 287 (1990), 703–718.CrossRefGoogle Scholar
[8]Kashiwara, M. and Vergne, M., ‘Functions on the Shilov boundary of the generalized half plane’, in: Noncommutative harmonic analysis (Proc. Third Colloq., Marseille-Luminy, 1978), Lect. Notes in Math. 728 (Springer, Berlin, 1979) pp. 136–176.Google Scholar
[9]Koecher, M., ‘Über eine Gruppe von rationalen Abbildungen’, Invent. Math. 3 (1967), 137–171.CrossRefGoogle Scholar
[10]Neher, E., ‘Cartan-Involutionen von halbeinfachen Jordan-Tripelsysteme’, Math. Z 169 (1979), 271–292.CrossRefGoogle Scholar
[11]Rubenthaler, H., ‘Une série dégénérée de représentations de SLn ⇈’, in: Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II (Springer, Berlin, 1979) pp. 427–459.Google Scholar
[12]Sahi, S., ‘Jordan algebras and degenerate principal series’, J. ReineAngew. Math. 462 (1995), 1–18.Google Scholar
[13]Sahi, S. and Stein, E. M., ‘Analysis in matrix space and Speh's representations’, Invent. Math. 101 (1990), 379–393.CrossRefGoogle Scholar
[14]Satake, I., Algebraic structures of symmetric domains, Kanô Memorial Lect. 4 (Iwanarni Shoten, Tokyo; Princeton Univ. Press, Princeton, NJ, 1980).Google Scholar
[15]Strichartz, R. S., ‘Fourier transform and non-compact rotation groups’, Indiana Univ. Math. J. 24 (1974/1975), 499–526.CrossRefGoogle Scholar
[16]Weil, A., ‘Sur certains groupes d'opérateurs unitaires’, Acta Math. 111 (1964), 143–211.CrossRefGoogle Scholar
[17]Zhang, G., ‘Jordan algebras and generalized principal series representations’, Math. Ann. 302 (1995), 773–786.CrossRefGoogle Scholar