Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n)$ and $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$ for principal series representations

TL;DR

This paper investigates Rankin-Selberg integrals for principal series representations of general linear groups over local fields, establishing explicit relations between different integral constructions and their invariance properties.

Contribution

It demonstrates that integrals over specific open orbits match Rankin-Selberg integrals up to an explicit constant, extending results to both $ ext{GL}_n imes ext{GL}_{n-1}$ and $ ext{GL}_n imes ext{GL}_n$ cases.

Findings

Equivalence of orbit integrals and Rankin-Selberg integrals with explicit constants

Extension of results to $ ext{GL}_n imes ext{GL}_{n-1}$ and $ ext{GL}_n imes ext{GL}_n$

Explicit formulas for constants relating different integral constructions

Abstract

Let $\mathsf k$ be a local field. Let $I_\nu$ and $I_{\nu'}$ be smooth principal series representations of $\mathrm{GL}_n(\mathsf k)$ and $\mathrm{GL}_{n-1}(\mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_\nu\times I_{\nu'}\rightarrow \mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_\nu\times I_{\nu'}\rightarrow \mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $\mathrm{GL}_n(\mathsf k)\times \mathrm{GL}_n(\mathsf k)$.