## Algebra & Number Theory

### Jordan blocks of cuspidal representations of symplectic groups

#### 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.

#### Article information

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

Dates
Revised: 2 June 2018
Accepted: 20 July 2018
First available in Project Euclid: 14 February 2019

https://projecteuclid.org/euclid.ant/1550113225

Digital Object Identifier
doi:10.2140/ant.2018.12.2327

Mathematical Reviews number (MathSciNet)
MR3911133

Zentralblatt MATH identifier
07026820

#### Citation

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. https://projecteuclid.org/euclid.ant/1550113225

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