Weyl’s law for the cuspidal spectrum of SLn

Abstract

Let r be a principal congruence subgroup of SL n (Z) and let σ be an irreducible unitary representation of SO(n). Let N Γ cus (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 a. In this paper we prove that the counting function N Γ cus (λ,σ) satisfies Weyl's law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SL n (Z).