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A cubic analogue of the cuspidal theta representations. (English) Zbl 0563.10022

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

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
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