Comptes Rendus
no. 5
Number Theory
Weyl's law for the cuspidal spectrum of SLn
[Une formule de Weyl pour le spectre cuspidal de SLn]

Présenté par : Jean-Michel Bismut

Werner Müller1
1 Universität Bonn, Mathematisches Institut, Beringstrasse 1, 53115 Bonn, Germany
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 347-352.

Soit Γ un sous-groupe de congruence principal de SL n (ℤ) et soit σ une représentation irréductible unitaire de SO(n). Soit NcusΓ(λ,σ) la fonction de dénombrement des valeurs propres de l'opérateur de Casimir, agissant sur l'espace des formes automorphes cuspidales pour Γ qui se transforment sous SO(n) par σ. Dans cette Note, nous prouvons une formule de Weyl pour le comportement asymptotique de la fonction de comptage NcusΓ(λ,σ).

Let Γ be a principal congruence subgroup of SL n (ℤ) and let σ be an irreducible unitary representation of SO(n). Let NcusΓ(λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to σ. In this Note we prove that the counting function NcusΓ(λ,σ) satisfies Weyl's law. In particular, this implies that there exist infinitely many cusp forms for the full modular group SL n (ℤ).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.01.003
Werner Müller 1

1 Universität Bonn, Mathematisches Institut, Beringstrasse 1, 53115 Bonn, Germany 
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Werner Müller. Weyl's law for the cuspidal spectrum of SLn. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 347-352. doi : 10.1016/j.crma.2004.01.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.003/

[1] J. Arthur A trace formula for reductive groups I: terms associated to classes in G(ℚ), Duke. Math. J., Volume 45 (1978), pp. 911-952

[2] J. Arthur The trace formula in invariant form, Ann. Math., Volume 114 (1981), pp. 1-74

[3] J. Arthur Intertwining operators and residues. I. Weighted characters, J. Funct. Anal., Volume 84 (1989), pp. 19-84

[4] J. Arthur On a family of distributions obtained from orbits, Canad. J. Math., Volume 38 (1986), pp. 179-214

[5] H. Donnelly On the cuspidal spectrum for finite volume symmetric spaces, J. Differential Geom., Volume 17 (1982), pp. 239-253

[6] W. Feller An Introduction to Probability Theory and its Applications, vol. II, Wiley, New York, 1971

[7] St. Miller On the existence and temperedness of cusp forms for SL 3 (ℤ), J. Reine Angew. Math., Volume 533 (2001), pp. 127-169

[8] C. Moeglin; J.-L. Waldspurger Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4), Volume 22 (1989), pp. 605-674

[9] W. Müller Eigenvalue estimates for locally symmetric spaces of finite volume, Symposium Partial Differential Equations, Holzhau, 1988, Teubner-Texte Math., vol. 112, Teubner, Leipzig, 1989, pp. 179-196

[10] W. Müller On the spectral side of the Arthur trace formula, Geom. Funct. Anal., Volume 12 (2002), pp. 669-722

[11] W. Müller, B. Speh, With appendix by E. Lapid, Absolute convergence of the spectral side of the Arthur trace formula for GLn, Geom. Funct. Anal., in press

[12] P. Sarnak On cusp forms (D. Hejhal et al., eds.), The Selberg Trace Formula and Related Topics, Contemp. Math., vol. 53, Amer. Math. Soc, 1984, pp. 393-407

[13] A. Selberg Harmonic analysis, Collected Papers, vol. I, Springer-Verlag, Berlin, 1989, pp. 626-674

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