The Weyl law for congruence subgroups and arbitrary $K_\infty$-types

Abstract

Let $G$ be a reductive algebraic group over $\mathbb{Q}$ and $\Gamma\subset G(\mathbb{Q})$ an arithmetic subgroup. Let $K_\infty\subset G(\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of $G(\mathbb{R})$ in $L^2(\Gamma\backslash G(\mathbb{R}))$ of a fixed $K_\infty$-type $\sigma$. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type $\sigma$ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about $L$-functions occurring in the constant terms of Eisenstein series. If $G$ satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.