Let
be a symplectic group over a nonarchimedean local field of characteristic zero and
odd residual characteristic. Given an irreducible cuspidal representation
of ,
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 ,
giving a bijection between the set of
endoparameters
for
and the set of restrictions to wild inertia of discrete Langlands parameters
for ,
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 ,
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