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 GL n (F) and n-dimensional ℓ-modular Deligne representations of the Weil group 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 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.
Highlights
Let F be a non-archimedean local field with finite residue field of cardinality q, a power of a prime p, and WF the Weil group of F
The -modular local Langlands correspondence established by Vignéras in [7] is a bijection from isomorphism classes of smooth irreducible representations of GLn(F ) and n-dimensional Deligne representations (Section 2.1) of the Weil group WF with nilpotent monodromy operator
It is uniquely characterized by a non-naive compatibility with the -adic local Langlands correspondence ( [1, 2, 5, 6]) under reduction modulo, involving twists by Zelevinsky involutions
Summary
Let F be a non-archimedean local field with finite residue field of cardinality q, a power of a prime p, and WF the Weil group of F. The -modular local Langlands correspondence established by Vignéras in [7] is a bijection from isomorphism classes of smooth irreducible representations of GLn(F ) and n-dimensional Deligne representations (Section 2.1) of the Weil group WF with nilpotent monodromy operator. It is uniquely characterized by a non-naive compatibility with the -adic local Langlands correspondence ( [1, 2, 5, 6]) under reduction modulo , involving twists by Zelevinsky involutions.
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