Relations between cusp forms sharing Hecke eigenvalues

Abstract

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n Baseline left-parenthesis double-struck upper A Subscript upper F Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mi>F</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_n(\mathbb {A}_F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is contained in the set of Hecke eigenvalues of a cusp form on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript m Baseline left-parenthesis double-struck upper A Subscript upper F Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mi>F</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_m(\mathbb {A}_F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to m"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \leq m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.