#### 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 $G×{GL}_{n}$, 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
Primary: 22E50
Secondary: 11F70