On Existence of Generic Cusp Forms on Semisimple Algebraic Groups

TL;DR

This paper investigates the existence of generic cuspidal automorphic representations with non-zero Fourier coefficients for semisimple algebraic groups over number fields, extending previous results to broader classes of groups.

Contribution

It generalizes known existence results of generic cuspidal automorphic representations to non-compact semisimple groups over number fields, including those that are quasi-split.

Findings

Existence of generic cuspidal automorphic representations for certain semisimple groups.

Extension of prior results to groups with non-compact Archimedean components.

Discussion on Fourier coefficients for automorphic forms on congruence subgroups.

Abstract

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for general semisimple algebraic group $G$ defined over a number field $k$ such that its Archimedean group $G_\infty$ is not compact. When $G$ is quasi--split over $k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize a result of Vign\' eras, Henniart, and Shahidi. We also discuss the existence of cuspidal automorphic forms with non--zero Fourier coefficients for congruence of subgroups of $G_\infty$.