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

Abstract

Let $F$ be a non archimedean local field of residual characteristic $p$ and $\ell $ a prime number different from $p$. Let $\mathrm{V}$ denote Vignéras’ $\ell $-modular local Langlands correspondence [7], between irreducible $\ell $-modular representations of $\mathrm{GL}_n(F)$ and $n$-dimensional $\ell $-modular Deligne representations of the Weil group $\mathrm{W}_F$. 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 $\mathrm{C}$, compatible with the formation of local constants in the generic case. In this note, following a remark of Alberto Mínguez, we characterize the modification $\mathrm{C}\circ \mathrm{V}^{-1}$ by a short list of natural properties.