Rankin–Selberg periods for spherical principal series

Jan Frahm, Feng Su

Research output: Contribution to journalArticlepeer-review

Abstract

By the unfolding method, Rankin–Selberg L-functions for GL(n)×GL(n′) 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 GL(n,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.

Original languageEnglish
Pages (from-to)1-33
Number of pages33
JournalManuscripta Mathematica
Volume168
Issue number1-2
DOIs
StatePublished - May 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

Funding

We thank Binyong Sun for sharing his insights and ideas on invariant functionals and for bringing the paper [] to our attention. We are very grateful to the referee for many helpful suggestions and remarks. The first author was supported by a research grant from the Villum Foundation (Grant No. 00025373). The second author was partly supported by the National Natural Science Foundation of China (No. 11901466) and the XJTLU Research Development Funding (RDF-19-02-04).

FundersFunder number
XJTLU Research Development FundingRDF-19-02-04
Villum Fonden00025373
National Natural Science Foundation of China11901466

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