An exceptional Siegel-Weil formula and poles of the Spin L-function of $PGSp_6$

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 $PGSp_6$ discovered by A. Pollack, we prove that a cuspidal representation of $PGSp_6$ is a (weak) functorial lift from the exceptional group $G_2$ if its (partial) Spin L-function has a pole at $s=1$.