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

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$s=1$.