Limit multiplicities for principal congruence subgroups of GL(n) and SL(n)

TL;DR

This paper investigates the asymptotic distribution of spectra for principal congruence subgroups of GL(n) and SL(n), demonstrating convergence to the Plancherel measure and advancing understanding of spectral limits in non-compact cases.

Contribution

It proves the limit multiplicity property for GL(n) and SL(n) and reduces the problem for general reductive groups using refined trace formula techniques.

Findings

Spectra of principal congruence subgroups converge to Plancherel measure.

Continuous spectrum contribution becomes negligible in the limit.

Main results extend known cases to unbounded rank groups.

Abstract

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups GL(n) and SL(n) we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur's trace formula obtained in [FLM11, FL11], which allows us to show that for GL(n) and SL(n) the contribution of the continuous spectrum is negligible in the limit.