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An exceptional Siegel–Weil formula and poles of the Spin L-function of $\text{PGSp}_{6}$

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  29 May 2020

Wee Teck Gan  and
Gordan Savin
Wee Teck Gan
Affiliation:
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore email matgwt@nus.edu.sg
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA email savin@math.utah.edu
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Abstract

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Keywords

Siegel–Weil formulaSpin L-functiontheta correspondence

MSC classification

Primary: 11F27: Theta series; Weil representation; theta correspondences 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 6 , June 2020 , pp. 1231 - 1261
Copyright
© The Authors 2020

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