Jordan blocks of cuspidal representations of symplectic groups

Blondel, Corinne; Henniart, Guy; Stevens, Shaun

doi:10.2140/ant.2018.12.2327

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Jordan blocks of cuspidal representations of symplectic groups

Corinne Blondel, Guy Henniart and Shaun Stevens

Vol. 12 (2018), No. 10, 2327–2386

DOI: 10.2140/ant.2018.12.2327

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 $G \times {GL}_{n}$ , 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

Vol. 12, No. 10, 2018