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

DOI10.1016/J.JFA.2024.110444arXiv2302.02207OpenAlexW4394742511MaRDI QIDQ6425428

Werner G. Müller

Publication date: 4 February 2023

Abstract: Let $G$ be a reductive algebraic group over $mathbbQ$ and $GammasubsetG(mathbbQ)$ an arithmetic subgroup. Let $K_{i}nftysubsetG(mathbbR)$ 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 $G(mathbbR)$ in $$ of a fixed $K_{i}nfty$-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.

Full work available at URL: https://doi.org/10.1016/j.jfa.2024.110444

Mathematics Subject Classification ID

Spectral theory; trace formulas (e.g., that of Selberg) (11F72) Heat and other parabolic equation methods for PDEs on manifolds (58J35)

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