A converse theorem for \(GL_ 4\)
Suppose \(k\) is a global field, \(\mathbb{A}\) the adeles over \(k\). The authors sketch the proofs of two converse theorems for \(GL_4(\mathbb{A})\). The first of these converse theorems can be stated as follows. Assume \(\pi\) is an irreducible admissible generic representation of \(GL_4 (\mathbb{A})\) with a central character trivial on the principal ideles \(k^\times\). Assume that \(\tau\) is an automorphic cuspidal representation of \(GL_m (\mathbb{A})\), for \(m=1,2\). Assume that for every \(\tau\) the function \(L(\pi \times\tau,s)\) is nice, meaning that it and the \(L\)-function for \(\widetilde \pi\times \widetilde\tau\) have analytic continuation to entire functions of \(s\), bounded in any vertical strip and satisfying a functional equation. Then the conclusion of the first converse theorem is that \(\pi\) is a cuspidal automorphic representation of \(GL_4(\mathbb{A})\). The second theorem excludes \(\tau\) unramified at a finite set \(S\) of places. The conclusion is that \(\pi_v\simeq\pi_v'\), for \(v\notin S\), where \(\pi'\) is an automorphic representation of \(GL_4 (\mathbb{A})\).
irreducible admissible generic representation
scientific article; zbMATH DE number 913096