Weyl's law for the cuspidal spectrum of SL n

Abstract

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) . To cite this article: W. Müller, C. R. Acad. Sci. Paris, Ser. I 338 (2004).