Rankin-Selberg periods for spherical principal series

TL;DR

This paper constructs invariant forms for spherical principal series of GL(n,R) to relate period integrals to L-functions, conjecturing their special values match local factors, verified in specific cases.

Contribution

It develops meromorphic families of invariant forms for spherical principal series and conjectures their special values relate to local L-factors, advancing the Bernstein-Reznikov approach.

Findings

Constructed meromorphic families of invariant forms for spherical principal series.

Conjectured that special values match archimedean local L-factors.

Verified the conjecture in several specific cases.

Abstract

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ and conjecture that their special values at the spherical vectors agree in absolute value with the archimedean local L-factors of the corresponding L-functions. We verify this conjecture in several cases. This work can be viewed as the first of two steps in a technique due to Bernstein-Reznikov for estimating L-functions using their period integral expressions.