Weyl's law for the cuspidal spectrum of SL<sub><i>n</i></sub>

Werner Müller

doi:10.1016/j.crma.2004.01.003

Number Theory

Weyl's law for the cuspidal spectrum of SL_n
[Une formule de Weyl pour le spectre cuspidal de SL_n]

Présenté par : Jean-Michel Bismut

Werner Müller ¹

¹ Universität Bonn, Mathematisches Institut, Beringstrasse 1, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 347-352.

Abstract (VO)
Résumé

Let Γ be a principal congruence subgroup of ${SL}_{n} (ℤ)$ and let σ be an irreducible unitary representation of SO(n). Let N_cus^Γ(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 N_cus^Γ(λ,σ) satisfies Weyl's law. In particular, this implies that there exist infinitely many cusp forms for the full modular group ${SL}_{n} (ℤ)$ .

Soit Γ un sous-groupe de congruence principal de ${SL}_{n} (ℤ)$ et soit σ une représentation irréductible unitaire de SO(n). Soit N_cus^Γ(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 N_cus^Γ(λ,σ).

Reçu le : 2003-10-21
Accepté le : 2004-01-05
Publié le : 2004-02-06

DOI : 10.1016/j.crma.2004.01.003

Affiliations des auteurs :

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 SL_n. 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/

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