Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n)

TL;DR

This paper establishes Weyl's law with an estimated remainder term for the cuspidal spectrum of arithmetic quotients of SL(n,R)/SO(n), extending spectral asymptotics to non-compact symmetric spaces.

Contribution

It extends the spectral asymptotics results of compact spaces to non-compact arithmetic quotients of SL(n,R)/SO(n).

Findings

Weyl's law derived for the cuspidal spectrum.

Estimated the remainder term in the spectral asymptotics.

Extended previous results to non-compact symmetric spaces.

Abstract

In this paper we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space S=SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends results of Duistermaat-Kolk-Varadarajan on spectral asymptotics for compact locally symmetric spaces to this non-compact setting.