Stable spectrum for pseudo-Riemannian locally symmetric spaces

Fanny Kassel; Toshiyuki Kobayashi

doi:10.1016/j.crma.2010.11.023

Group Theory/Harmonic Analysis

Stable spectrum for pseudo-Riemannian locally symmetric spaces
[Spectre stable pour les variétés pseudo-riemanniennes localement symétriques]

Présenté par : Michèle Vergne

Fanny Kassel ¹ ; Toshiyuki Kobayashi ²

¹ Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA
² Graduate School of Mathematical Sciences, IPMU, The University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914 Japan

Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 29-33.

Résumé
Abstract

Soit $X = G / H$ un espace symétrique réductif vérifiant $rang G / H = rang K / K \cap H$ , où K (resp. $K \cap H$ ) est un sous-groupe compact maximal de G (resp. de H). Nous étudions le spectre discret de certaines formes de Clifford–Klein $Γ \ X$ , où Γ est un sous-groupe discret de G agissant librement et proprement sur X : nous construisons un ensemble infini de valeurs propres pour les opérateurs différentiels « intrinsèques » sur $Γ \ X$ , et cet ensemble est stable par petites déformations de Γ dans G.

Let $X = G / H$ be a reductive symmetric space with $rank G / H = rank K / K \cap H$ , where K (resp. $K \cap H$ ) is a maximal compact subgroup of G (resp. of H). We investigate the discrete spectrum of certain Clifford–Klein forms $Γ \ X$ , where Γ is a discrete subgroup of G acting properly discontinuously and freely on X: we construct an infinite set of joint eigenvalues for “intrisic” differential operators on $Γ \ X$ , and this set is stable under small deformations of Γ in G.

Reçu le : 2010-06-23
Accepté le : 2010-11-22
Publié le : 2010-12-24

DOI : 10.1016/j.crma.2010.11.023

Affiliations des auteurs :

Fanny Kassel ¹ ; Toshiyuki Kobayashi ²

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     author = {Fanny Kassel and Toshiyuki Kobayashi},
     title = {Stable spectrum for {pseudo-Riemannian} locally symmetric spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {29--33},
     publisher = {Elsevier},
     volume = {349},
     number = {1-2},
     year = {2011},
     doi = {10.1016/j.crma.2010.11.023},
     language = {en},
}

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%A Toshiyuki Kobayashi
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Fanny Kassel; Toshiyuki Kobayashi. Stable spectrum for pseudo-Riemannian locally symmetric spaces. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 29-33. doi : 10.1016/j.crma.2010.11.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.023/

Bibliographie
Cité par

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[3] O. Guichard Groupes plongés quasi-isométriquement dans un groupe de Lie, Math. Ann., Volume 330 (2004), pp. 331-351

[4] S. Helgason Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000

[5] F. Kassel Deformation of proper actions on reductive homogeneous spaces | arXiv

[6] F. Kassel, Quotients compacts d'espaces homogènes réels ou p-adiques, PhD thesis, Université Paris-Sud 11, November 2009, see http://www.math.u-psud.fr/~kassel/.

[7] B. Klingler Complétude des variétés lorentziennes à courbure constante, Math. Ann., Volume 306 (1996), pp. 353-370

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[10] T. Kobayashi Deformation of compact Clifford–Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann., Volume 310 (1998), pp. 394-408

[11] T. Kobayashi Hidden symmetries and spectrum of the Laplacian on an indefinite Riemannian manifold, Spectral Analysis in Geometry and Number Theory, Contemp. Math., vol. 484, American Mathematical Society, Providence, RI, 2009, pp. 73-87

[12] T. Kobayashi; T. Yoshino Compact Clifford–Klein forms of symmetric spaces — revisited, Pure Appl. Math. Quart., Volume 1 (2005), pp. 591-653

[13] R.S. Kulkarni; F. Raymond 3-Dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom., Volume 21 (1985), pp. 231-268

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