Weyl's law for the cuspidal spectrum of SLn
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. Muller, C. R. Acad. Sci. Paris, Ser. I 338 (2004).