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Branching laws for classical groups: the non-tempered case

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  17 December 2020

Wee Teck Gan ,
Benedict H. Gross  and
Dipendra Prasad
Wee Teck Gan
Affiliation:
National University of Singapore, Singapore119076matgwt@nus.edu.sg
Benedict H. Gross
Affiliation:
Department of Mathematics, University of California San Diego, La Jolla, CA92093, USAgross@math.harvard.edu
Dipendra Prasad
Affiliation:
Indian Institute of Technology Bombay, Powai, Mumbai400076, Indiaprasad.dipendra@gmail.com St Petersburg State University, St Petersburg, Russia
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Abstract

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Keywords

Gan–Gross–Prasad conjecturesL-packetsA-packetsL-parametersA-parametersrelevant pair of A-parameterslocal branching lawsperiod integralscentral L-valuesclassical groupsL-functionsepsilon factorsrepresentation theory

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 11 , November 2020 , pp. 2298 - 2367
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Copyright
© The Author(s) 2020

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Footnotes

WTG is partially supported by an MOE Tier 2 grant R146-000-233-112. DP thanks the Science and Engineering Research Board of the Department of Science and Technology, India for its support through the JC Bose National Fellowship of the Government of India, project number JBR/2020/000006. His work was also supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the agreement 14.W03.31.0030 dated 15 February 2018.

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