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ON FORMAL DEGREES OF UNIPOTENT REPRESENTATIONS

Part of: Lie groups Linear algebraic groups and related topics Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  19 March 2021

Yongqi Feng ,
Eric Opdam  and
Maarten Solleveld Open the ORCID record for Maarten Solleveld [Opens in a new window]
Yongqi Feng
Affiliation:
Department of Mathematics, Shantou University, Daxue Road 243, 515063 Shantou, China (yqfeng@stu.edu.cn)
Eric Opdam
Affiliation:
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands (e.m.opdam@uva.nl)
Maarten Solleveld
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands (m.solleveld@science.ru.nl)
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Abstract

Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation $\pi $ can be expressed in terms of the adjoint $\gamma $ -factor of the enhanced L-parameter of $\pi $ . A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.

We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $\gamma $ -factors.

Keywords

representation theoryp-adic groupsunipotent representationsformal degrees

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Secondary: 11S37: Langlands-Weil conjectures, nonabelian class field theory 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 1947 - 1999
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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