Relations between cusp forms sharing Hecke eigenvalues

TL;DR

This paper investigates the conditions under which the Hecke eigenvalues of cusp forms on different general linear groups are related, connecting automorphic forms with finite-dimensional group representations.

Contribution

It explores the inclusion relations of Hecke eigenvalues between cusp forms on different GL groups and links this to finite-dimensional group representations, providing new insights.

Findings

Characterization of when Hecke eigenvalues of cusp forms are contained in those of larger groups

Connection established between cusp form eigenvalues and finite-dimensional representations

Results applicable to automorphic forms on GL_n and GL_m

Abstract

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on $GL_n(A_F)$ is contained in the set of Hecke eigenvalues of a cusp form on $GL_m(A_F)$ for $n \leq m$.This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.