Patterson, S. J.; Piatetski-Shapiro, Ilya I. A cubic analogue of the cuspidal theta representations. (English) Zbl 0563.10022 J. Math. Pures Appl., IX. Sér. 63, 333-375 (1984). Let \(C_ F\) denote either the multiplicative group of a local field or the idèle class group of a global field. Five years ago, the second author and the reviewer constructed a correspondence \(\chi\) \(\to r(\chi)\) from the set of quasicharacters of \(C_ F\) to the set of irreducible ”distinguished” representations of the metaplectic covering group \(\tilde G_ 2\) of \(GL_ 2(F)\); in case F is global, each r(\(\chi)\) is an automorphic representation of \(\tilde G;\) in fact, an \(r(\chi_ v)\) is supercuspidal if and only if \(\chi_ v\) is not of the form \(\mu^ 2\), and r(\(\chi)\) is cuspidal automorphic if and only if one local component is supercuspidal. In the present paper, the authors derive an analogous correspondence \(\chi\) \(\to r'(\chi)\) with \(\tilde G_ 2\) replaced by a three-fold metaplectic cover of \(GL_ 3\). The construction is complicated and relies on an ingenious application of the converse theorem of Hecke theory for \(GL_ 3\), suitably modified for the metaplectic cover. The end result is (roughly) that r’(\(\chi)\) is cuspidal if and only if \(\chi\) is not a cube. Reviewer: S.Gelbart Cited in 1 ReviewCited in 7 Documents MSC: 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings Keywords:theta-representations; metaplectic groups; automorphic representation; three-fold metaplectic cover of \(GL_ 3\); converse theorem of Hecke theory; cuspidal PDFBibTeX XMLCite \textit{S. J. Patterson} and \textit{I. I. Piatetski-Shapiro}, J. Math. Pures Appl. (9) 63, 333--375 (1984; Zbl 0563.10022)