Irreducibility Criteria for Local and Global Representations

It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given. Introduction One of the motivations for this paper was to show that full level cuspidal Siegel eigenforms generate irreducible, automorphic representations of the adelic symplectic similitude group. Such a result is well known for the case of classical elliptic modular forms. Given an elliptic cusp form f (of some weight and some level), an adelic function Φf can be constructed (see [Ge], §5), which is a cuspidal automorphic form on the adelic group GL(2,A); here, A denotes the ring of adeles of Q. Let Vf be the space of automorphic forms generated by all right translates of Φf . In this classical situation it turns out that Vf is irreducible precisely when f is an eigenform for the Hecke operators Tp for almost all primes p. The proof uses the strong multiplicity one property for cuspidal automorphic representations of GL(2). For most other types of modular forms, strong multiplicity one, or even weak multiplicity one, is not available. The goal of this paper is to show that, under certain circumstances, the automorphic representation Vf is still irreducible, even if multiplicity one is not known. Loosely speaking, this is the case whenever f is a holomorphic type of modular form and is an eigenfunction for all Hecke operators. See Corollary 3.2 for a precise statement. We stress that this result does not prove multiplicity one for full level automorphic forms; for example, under our current state of knowledge, it is still conceivable that two holomorphic Siegel cusp forms of degree n > 1 have the same weight and the same Hecke eigenvalues for all primes p, yet are linearly independent. While our results are applicable mainly in a reductive setting, we keep the definitions general enough to include certain non-reductive situations, such as Jacobi forms. For completeness we also include analogous local Archimedean and nonArchimedean irreducibility criteria. All of this is well known to experts, but for Received by the editors June 6, 2011. 2010 Mathematics Subject Classification. Primary 11F46, 11F50, 11F70; Secondary 22E50, 22E55.