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

## TL;DR

This paper establishes a new Siegel-Weil formula in the context of exceptional theta correspondence and uses it alongside a novel Rankin-Selberg integral to analyze poles of the Spin L-function of PGSp_6, linking it to functorial lifts from G_2.

## Contribution

It introduces a new Siegel-Weil formula for exceptional theta correspondence and connects poles of the Spin L-function to functorial lifts from G_2.

## Key findings

- Proves a Siegel-Weil formula in the exceptional setting.
- Identifies conditions under which a representation lifts from G_2.
- Shows poles of the Spin L-function imply functorial lifts.

## 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 $s=1$.