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Endoscopic lifts from PGL3 to G2

Published online by Cambridge University Press:  04 December 2007

Wee Teck Gan  and
Gordan Savin
Wee Teck Gan
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USAwgan@math.ucsd.edu Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USAsavin@math.utah.edu
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Abstract

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We determine essentially completely the theta correspondence arising from the dual pair ${\it PGL}_3 \times G_2 \subset E_6$ over a p-adic field. Our first result determines the theta lift of any non-supercuspidal representation of PGL3 and shows that the lifting respects Langlands functoriality. Our second result shows that the theta lift $\theta(\pi)$ of a (non-self-dual) supercuspidal representation $\pi$ of PGL3 is an irreducible generic supercuspidal representation of G2; we also determine $\theta(\pi)$ explicitly when $\pi$ has depth zero.

Keywords

theta correspondencesupercuspidal lifts22E3522E50 (primary)11F70 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 3 , May 2004 , pp. 793 - 808
Copyright
Foundation Compositio Mathematica 2004