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

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

1 Scopus citations

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
State	Published - Oct 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023, Science China Press.

Funding

Dongwen Liu was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LZ22A010006) and National Natural Science Foundation of China (Grant No. 12171421). Feng Su was supported by National Natural Science Foundation of China (Grant No. 11901466) and the Qinglan Project of Jiangsu Province. Binyong Sun was supported by the National Key R&D Program of China (Grant No. 2020YFA0712600). The authors thank the referees for the careful reading and comments.

Funders	Funder number
National Natural Science Foundation of China	11901466, 12171421
Natural Science Foundation of Zhejiang Province	LZ22A010006
National Key Research and Development Program of China	2020YFA0712600
Qinglan Project of Jiangsu Province of China

Keywords

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

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10.1007/s11425-022-2050-5

Cite this

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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).",

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AU - Sun, Binyong

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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).

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

Abstract

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Keywords

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