Algebra & Number Theory

Jordan blocks of cuspidal representations of symplectic groups

Corinne Blondel, Guy Henniart, and Shaun Stevens

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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.

Article information

Algebra Number Theory, Volume 12, Number 10 (2018), 2327-2386.

Received: 30 May 2017
Revised: 2 June 2018
Accepted: 20 July 2018
First available in Project Euclid: 14 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

local Langlands correspondence symplectic group $p$-adic group Jordan block endoparameter types and covers


Blondel, Corinne; Henniart, Guy; Stevens, Shaun. Jordan blocks of cuspidal representations of symplectic groups. Algebra Number Theory 12 (2018), no. 10, 2327--2386. doi:10.2140/ant.2018.12.2327.

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