A characterization of the relation between two $\ell $-modular correspondences

Abstract

Let F be a non archimedean local field of residual characteristic p and ℓ a prime number different from p. Let V denote Vignéras’ ℓ-modular local Langlands correspondence [7], between irreducible ℓ-modular representations of GLn(F) and n-dimensional ℓ-modular Deligne representations of the Weil group WF. In [4], enlarging the space of Galois parameters to Deligne representations with non necessarily nilpotent operators allowed us to propose a modification of the correspondence of Vignéras into a correspondence C, compatible with the formation of local constants in the generic case. In this note, following a remark of Alberto Mínguez, we characterize the modification C∘V−1 by a short list of natural properties.