Stable spectrum for pseudo-Riemannian locally symmetric spaces

Kassel, Fanny; Kobayashi, Toshiyuki

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

Kassel, Fanny ¹ ; Kobayashi, Toshiyuki 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, Tome 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 :

Kassel, Fanny ¹ ; Kobayashi, Toshiyuki ²

@article{CRMATH_2011__349_1-2_29_0,
     author = {Kassel, Fanny and Kobayashi, Toshiyuki},
     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},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2010.11.023/}
}

TY  - JOUR
AU  - Kassel, Fanny
AU  - Kobayashi, Toshiyuki
TI  - Stable spectrum for pseudo-Riemannian locally symmetric spaces
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 29
EP  - 33
VL  - 349
IS  - 1-2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2010.11.023/
DO  - 10.1016/j.crma.2010.11.023
LA  - en
ID  - CRMATH_2011__349_1-2_29_0
ER  -

%0 Journal Article
%A Kassel, Fanny
%A Kobayashi, Toshiyuki
%T Stable spectrum for pseudo-Riemannian locally symmetric spaces
%J Comptes Rendus. Mathématique
%D 2011
%P 29-33
%V 349
%N 1-2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2010.11.023/
%R 10.1016/j.crma.2010.11.023
%G en
%F CRMATH_2011__349_1-2_29_0

Kassel, Fanny; Kobayashi, Toshiyuki. Stable spectrum for pseudo-Riemannian locally symmetric spaces. Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 29-33. doi : 10.1016/j.crma.2010.11.023. http://archive.numdam.org/articles/10.1016/j.crma.2010.11.023/

Bibliographie
Cité par

[1] Benoist, Y. Actions propres sur les espaces homogènes réductifs, Ann. Math., Volume 144 (1996), pp. 315-347

[2] Flensted-Jensen, M. Discrete series for semisimple symmetric spaces, Ann. Math., Volume 111 (1980), pp. 253-311

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

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

[8] Kobayashi, T. Proper action on a homogeneous space of reductive type, Math. Ann., Volume 285 (1989), pp. 249-263

[9] Kobayashi, T. Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory, Volume 6 (1996), pp. 147-163

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

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

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

[14] Matsuki, T.; Oshima, T. A description of discrete series for semisimple symmetric spaces, Group Representations and Systems of Differential Equations, Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 331-390

Cité par Sources :