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

TL;DR

This paper proves a version of Weyl's law for the spectrum of automorphic forms on certain algebraic groups, confirming a conjecture by Sarnak for groups satisfying a specific property.

Contribution

It establishes Sarnak's conjecture on Weyl's law for the spectrum of automorphic forms for groups with property (L), extending results to classical groups over number fields.

Findings

Weyl's law holds for the cuspidal spectrum of fixed K-infinity type.

Residual spectrum grows at a lower order than the cuspidal spectrum.

The conjecture is proved for groups satisfying property (L).

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.