Comptes Rendus
no. 5
Number Theory
Weyl's law for the cuspidal spectrum of SLn

Presented by: 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.

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).

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).

Received:
Accepted:
Published online:
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/

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