Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations

Jian Shu Li; Dongwen Liu; Feng Su; Binyong Sun

doi:10.1007/s11425-022-2050-5

Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations

Jian Shu Li, Dongwen Liu, Feng Su^*, Binyong Sun

^*Corresponding author for this work

Department of Pure Mathematics

Zhejiang University

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let k be a local field. Let I_v and Iv′ be smooth principal series representations of GL_n(k) and GL_n-−1(k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map Iv×Iv′→C with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map Iv×Iv′→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 GL_n(k) × GL_n(k).

Original language	English
Pages (from-to)	2203-2218
Number of pages	16
Journal	Science China Mathematics
Volume	66
Issue number	10
DOIs	https://doi.org/10.1007/s11425-022-2050-5
Publication status	Published - Oct 2023

Keywords

22E50
43A80
L-functions
Rankin-Selberg convolutions
principal series representations

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10.1007/s11425-022-2050-5

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title = "Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations",

abstract = "Let k be a local field. Let Iv and Iv′ be smooth principal series representations of GLn(k) and GLn-−1(k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map Iv×Iv′→C with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map Iv×Iv′→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 GLn(k) × GLn(k).",

keywords = "22E50, 43A80, L-functions, Rankin-Selberg convolutions, principal series representations",

author = "Li, {Jian Shu} and Dongwen Liu and Feng Su and Binyong Sun",

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AU - Li, Jian Shu

AU - Liu, Dongwen

AU - Su, Feng

AU - Sun, Binyong

PY - 2023/10

Y1 - 2023/10

N2 - Let k be a local field. Let Iv and Iv′ be smooth principal series representations of GLn(k) and GLn-−1(k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map Iv×Iv′→C with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map Iv×Iv′→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 GLn(k) × GLn(k).

AB - Let k be a local field. Let Iv and Iv′ be smooth principal series representations of GLn(k) and GLn-−1(k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map Iv×Iv′→C with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map Iv×Iv′→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 GLn(k) × GLn(k).

KW - 22E50

KW - 43A80

KW - L-functions

KW - Rankin-Selberg convolutions

KW - principal series representations

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SN - 1674-7283

VL - 66

SP - 2203

EP - 2218

JO - Science China Mathematics

JF - Science China Mathematics

IS - 10

ER -

Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations

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