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 representation of SO(n). Let $N(T,σ)$ 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 $σ$. We prove that the counting function $N(T,σ)$ satisfies Weyl's law as $T\to\infty$. Especially this implies that there exist infinitely many cusp forms for the full modular group $SL_n(Z)$.