Arthur parameters and cuspidal automorphic modules of classical groups

Dihua Jiang; Lei Zhang

doi:10.4007/annals.2020.191.3.2

Arthur parameters and cuspidal automorphic modules of classical groups

Dihua Jiang, Lei Zhang

School of Mathematics

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

14 Scopus citations

Abstract

The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

Original language	English (US)
Pages (from-to)	739-827
Number of pages	89
Journal	Annals of Mathematics
Volume	191
Issue number	3
DOIs	https://doi.org/10.4007/annals.2020.191.3.2
State	Published - May 2020

Bibliographical note

Funding Information:
Keywords: Arthur parameters, cuspidal automorphic modules, Bessel-Fourier coefficients, twisted automorphic descent, classical groups, global Gan-Gross-Prasad conjecture AMS Classification: Primary: 11F70, 22E50; Secondary: 11F85, 22E55. The research of the first named author is supported in part by the NSF Grants DMS-1301567, DMS-1600685 and DMS-1901802; that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore. ©c 2020 Department of Mathematics, Princeton University.

Publisher Copyright:
© 2020. Department of Mathematics, Princeton University.

Keywords

Arthur parameters
Bessel-fourier coefficients
Classical groups
Cuspidal automorphic modules
Global gan-gross-prasad conjecture
Twisted automorphic descent

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title = "Arthur parameters and cuspidal automorphic modules of classical groups",

abstract = "The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.",

keywords = "Arthur parameters, Bessel-fourier coefficients, Classical groups, Cuspidal automorphic modules, Global gan-gross-prasad conjecture, Twisted automorphic descent",

author = "Dihua Jiang and Lei Zhang",

note = "Funding Information: Keywords: Arthur parameters, cuspidal automorphic modules, Bessel-Fourier coefficients, twisted automorphic descent, classical groups, global Gan-Gross-Prasad conjecture AMS Classification: Primary: 11F70, 22E50; Secondary: 11F85, 22E55. The research of the first named author is supported in part by the NSF Grants DMS-1301567, DMS-1600685 and DMS-1901802; that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore. {\textcopyright}c 2020 Department of Mathematics, Princeton University. Publisher Copyright: {\textcopyright} 2020. Department of Mathematics, Princeton University.",

year = "2020",

month = may,

doi = "10.4007/annals.2020.191.3.2",

language = "English (US)",

volume = "191",

pages = "739--827",

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issn = "0003-486X",

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AU - Jiang, Dihua

AU - Zhang, Lei

N1 - Funding Information: Keywords: Arthur parameters, cuspidal automorphic modules, Bessel-Fourier coefficients, twisted automorphic descent, classical groups, global Gan-Gross-Prasad conjecture AMS Classification: Primary: 11F70, 22E50; Secondary: 11F85, 22E55. The research of the first named author is supported in part by the NSF Grants DMS-1301567, DMS-1600685 and DMS-1901802; that of the second named author is supported in part by the start-up grant, AcRF Tier 1 grants R-146-000-237-114 and R-146-000-277-114 of National University of Singapore. ©c 2020 Department of Mathematics, Princeton University. Publisher Copyright: © 2020. Department of Mathematics, Princeton University.

PY - 2020/5

Y1 - 2020/5

N2 - The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

AB - The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

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KW - Twisted automorphic descent

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Arthur parameters and cuspidal automorphic modules of classical groups

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