Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ attached to Newforms and Oldforms

TL;DR

This paper computes Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ representations associated with newforms and oldforms, advancing the understanding of their structure under the conjectural theory.

Contribution

It provides explicit calculations of integrals attached to newforms and oldforms for generic representations of $p$-adic ${\rm SO}_{2n+1}$, assuming the conjectural newform theory.

Findings

Explicit formulas for Rankin-Selberg integrals for newforms.

Verification of conjectural properties for supercuspidal cases.

Extension of integral computations to oldforms.

Abstract

The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.