On Automorphic Forms on GL 2 and Hecke Operators

Abstract

In [4], Jacquet and Langlands showed that if an irreducible unitary representation of the general linear group of degree 2 over the adele ring of an A-field' is an irreducible constituent of the representation on the space of cuspidal functions, then it is decomposable, i.e., it is a tensor product of irreducible representations of local components of the group. In the present paper, we prove the semi-simplicity of the algebra generated by some Hecke operators operating on a certain space of cusp forms. As a consequence of this, we show that two such irreducible constituents are equivalent if their local factors are equivalent for almost all places including all archimedean places. In more detail, let F be an A-field, and FA (resp. FAX) the adele ring (resp. the idele group) of F. We put GF= GL2(F) and GA = GL2(FA). We know that the center ZA of GA is isomorphic to FAX. For a unitary character X of the idele class group FA /Fs, let L'(GF\GA, X) denote the space of measurable functions on GA satisfying