Vol. 12, No. 10, 2018

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Jordan blocks of cuspidal representations of symplectic groups

Corinne Blondel, Guy Henniart and Shaun Stevens

Vol. 12 (2018), No. 10, 2327–2386
Abstract

Let G be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Mœglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a ramification theorem for G, giving a bijection between the set of endoparameters for G and the set of restrictions to wild inertia of discrete Langlands parameters for G, compatible with the local Langlands correspondence. The main tool consists in analyzing the Hecke algebra of a good cover, in the sense of Bushnell–Kutzko, for parabolic induction from a cuspidal representation of G × GLn, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine reducibility points; a criterion of Mœglin then relates this to Langlands parameters.

Keywords
local Langlands correspondence, symplectic group, $p$-adic group, Jordan block, endoparameter, types and covers
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 11F70
Milestones
Received: 30 May 2017
Revised: 2 June 2018
Accepted: 20 July 2018
Published: 1 February 2019
Authors
Corinne Blondel
CNRS-IMJ-PRG
Université Paris Diderot
Paris
France
Guy Henniart
Université de Paris-Sud
Laboratoire de Mathématiques d’Orsay
Orsay
France
Shaun Stevens
School of Mathematics
University of East Anglia
Norwich
United Kingdom