Weyl’s law for the cuspidal spectrum of $\mathrm{SL}_n$

Abstract

Let $\Gamma$ be a principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ and let $\sigma$ be an irreducible unitary representation of $\mathrm{SO}(n)$. Let $N^\Gamma_{\mathrm{cu}}(\lambda,\sigma)$ be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for $\Gamma$ which transform under $\mathrm{SO}(n)$ according to $\sigma$. In this paper we prove that the counting function $N^\Gamma_{\mathrm{cu}}(\lambda,\sigma)$ satisfies Weyl’s law. Especially, this implies that there exist infinitely many cusp forms for the full modular group $\mathrm{SL}_n(\mathbb{Z})$.