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Dichotomy for generic supercuspidal representations of G2

Part of: Algebraic number theory: global fields Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 February 2011

Gordan Savin  and
Martin H. Weissman
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (email: savin@math.utah.edu)
Martin H. Weissman
Affiliation:
Department of Mathematics, University of California, Santa Cruz, CA 95064, USA (email: weissman@ucsc.edu)
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Abstract

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The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This correspondence arises from theta correspondences in E6 and E7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G2 to a single conjecture about the parameterization for PGSp 6.

Keywords

G2dichotomyLanglandslocal fieldp-adicShalikasupercuspidal

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11R39: Langlands-Weil conjectures, nonabelian class field theory 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 735 - 783
Copyright
Copyright © Foundation Compositio Mathematica 2011

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