On the Cusp Forms of Congruence Subgroups of an almost Simple Lie group

TL;DR

This paper investigates the existence of cusp forms for almost simple Lie groups, especially focusing on $SL_M(R)$, by combining representation theory and local $p$-adic group information to establish new existence results.

Contribution

It introduces new methods for proving the existence of cusp forms on almost simple Lie groups, with specific results for principal congruence subgroups of $SL_M(R)$.

Findings

Proved existence of cusp forms for certain congruence subgroups of $SL_M(R)$

Developed a combined approach using global and local representation theory

Established new criteria for the existence of cusp forms in this setting

Abstract

In this paper we address the issue of existence of cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for $p$-adic groups known by the first author. We pay special attention to the case of $SL_M(\Bbb R)$ where we prove various existence results for principal congruence subgroups.