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

Abstract

Let $\mathsf{k}$ be a local field. Let $I_{\nu }$ and $I_{{\nu }^{\prime }}$ 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 $I_{\nu }\times I_{{\nu }^{\prime }}\to \mathbb{C}$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_{\nu }\times I_{{\nu }^{\prime }}\to \mathbb{C}$ 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 ${\mathrm{GL}}_n(\mathsf{k})\times {\mathrm{GL}}_n(\mathsf{k})$.

References in corpus (2)

• Schwartz functions on Nash manifolds
• Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$