A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform

[Un problème de Cauchy pour des opérateurs différentiels elliptiques invariants et continuité d’une transformée de Berezin généralisée.]

Ørsted, Bent ; Vargas, Jorge

Annales de l'Institut Fourier, Tome 57 (2007), p. 693-702 / Harvested from Numdam

Résumé

Dans cette note, nous généralisons les résultats de notre article précédent sur l’opérateur de Casimir et sur la transformée de Berezin, en prouvant la continuité $(L^2,L^2)$ d’une transformée de Berezin généralisée associée à un problème de bifurcation pour une classe de représentations unitaires définie par des opérateurs elliptiques invariants. Nous prouvons aussi que, sous des conditions générales adéquates, cette transformée de Berezin généralisée est $(L^p, L^p)$-continue pour $ 1 \le p \le \infty .$

Abstract

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $(L^p, L^p)$-continuous for $ 1 \le p \le \infty .$

Détail

Publié le : 2007-01-01

MR 2336825 | Zbl 1123.22008

DOI : https://doi.org/10.5802/aif.2272
Classification:  22E46
Mots clés: Séries discrètes, lois de bifurcation, opérateurs elliptiques invariants.

Bibliographie

[1] Atiyah, M. M., Elliptic operators, discrete groups and von Neumann algebras, Astérisque, Tome 32-33 (1976), pp. 43-72 | MR 420729 | Zbl 0323.58015

[2] Connes, A.; Moscovici, H. The $L^2-$index theorem for homogeneous spaces of Lie groups, Ann. of Math. (2), Tome 115 (1982) no. 2, pp. 291-330 | Article | MR 647808 | Zbl 0515.58031

[3] Cowling, M. The Kunze-Stein phenomenon, Ann. of Math., Tome 107 (1978), pp. 209-234 | Article | MR 507240 | Zbl 0363.22007

[4] Jakobsen, H. P.; Vergne, M. Restriction and expansions of holomorphic representations, Journ. of Funct. Anal., Tome 34 (1979), pp. 29-53 | Article | MR 551108 | Zbl 0433.22011

[5] Knapp, A. Representation theory of Semisimple Lie Groups, An overview based in examples, Princeton Univ. Press, Princeton Mathematical Series (1986) | MR 855239 | Zbl 0604.22001

[6] Kobayashi, T. Harmonic Analysis on Homogeneous Manifolds of Reductive Type and Unitary Representation Theory, Selected Papers on harmonic Analysis, groups and invariants, K. Nomizu (Editor) (Amer. Math. Soc. Transl. Ser. 2) Tome 183 (1999), pp. 1-33

[7] Kobayashi, T. Discretely decomposable restrictions of unitary representations of reductive Lie groups - examples and conjectures, Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, T. Kobayashi (Editor), Tokyo (Advanced Studies in Pure Mathematics) Tome 26 (2000), pp. 99-127 (Kinokuniya) | MR 1770719 | Zbl 0959.22009

[8] Neeb, Karl-Hermann Holomorphy and Convexity in Lie Theory, Walter de Gruyter, De Gruyter expositions in mathematics, Tome 28 (2000) | MR 1740617 | Zbl 0936.22001

[9] Orsted, B.; Vargas, J. Restriction of square integrable representations: Discrete Spectrum, Duke Math. Journal, Tome 123 (2004), pp. 609-633 | Article | MR 2068970 | Zbl 1056.22008

[10] Treves, S. Topological vector spaces, distributions and kernels, Academic Press, New York (1967) | MR 225131 | Zbl 0171.10402

[11] Trombi, P.; Varadarajan, V. Asymptotic behavior of eigenfunctions on a semisimple Lie group: The Discrete Spectrum, Acta Math., Tome 129 (1972), pp. 237-280 | Article | MR 393349 | Zbl 0244.43006

[12] Wallach, N.; Wolf, J. Completeness of Poincare Series for Automorphic Forms Associated to the Integrable Discrete Series, Representation Theory of reductive groups, P. Trombi (editor) (Progress in Math.) Tome 40 (1983), pp. 265-281 | MR 733818 | Zbl 0566.22014