On the nonvanishing hypothesis for Rankin-Selberg convolutions for $\mathrm{GL}_n(\mathbb C)\times \mathrm{GL}_n(\mathbb C)$

TL;DR

This paper proves the nonvanishing hypothesis for Rankin-Selberg convolutions at the central critical point for the group pair $ ext{GL}_n( ext{C}) imes ext{GL}_n( ext{C})$, extending previous results to a new setting.

Contribution

It confirms the nonvanishing hypothesis for $ ext{GL}_n( ext{C}) imes ext{GL}_n( ext{C})$ at the central critical point, building on Sun's earlier work for related groups.

Findings

Established nonvanishing at the central critical point for $ ext{GL}_n( ext{C}) imes ext{GL}_n( ext{C})$

Extended the nonvanishing hypothesis to complex general linear groups

Confirmed conjectural behavior for these convolutions at the central point.

Abstract

Inspired by Sun's breakthrough in establishing the nonvanishing hypothesis for Rankin-Selberg convolutions for the groups $\mathrm{GL}_n (\mathbb R)\times \mathrm{GL}_{n-1} (\mathbb R)$ and $\mathrm{GL}_n (\mathbb C)\times \mathrm{GL}_{n-1} (\mathbb C)$, we confirm it for $\mathrm{GL}_{n} (\mathbb C)\times \mathrm{GL}_n (\mathbb C)$ at the central critical point.