Weyl's law for the cuspidal spectrum of SL(n)

Abstract

Let $\Gamma$ be a principal congruence subgroup of $SL_n(Z)$ and let $\sigma$ be an irreducible representation of SO(n). Let $N(T,\sigma)$ be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for $\Gamma$ which transform under SO(n) according to $\sigma$. We prove that the counting function $N(T,\sigma)$ satisfies Weyl's law as $T\to\infty$. Especially this implies that there exist infinitely many cusp forms for the full modular group $SL_n(Z)$.