Correspondance de Howe: paires duales de type II

TL;DR

This paper introduces a new method for establishing Howe correspondence for dual pairs of type (GL_n, GL_m) over non-Archimedean fields, combining Kudla's filtration and irreducibility results to explicitly describe the correspondence via Langlands parameters.

Contribution

It presents a novel approach that unifies previous techniques to prove Howe correspondence explicitly for dual pairs of type (GL_n, GL_m) over any characteristic non-Archimedean field.

Findings

Proof valid for any characteristic non-Archimedean fields

Explicit description of Howe correspondence via Langlands parameters

Method combines Kudla's filtration with irreducibility results

Abstract

In this article, we give a new method for proving Howe correspondence in the case of dual pairs of type $({\rm GL}_n, {\rm GL}_m)$ over a non-Archimedean locally compact field $F$. The proof consists in combining a study on Kudla's filtration with the results of a previous article of the autor about the irreducibility of a parabolically induced representation. The proof is valid for $F$ of any characteristic and allows us to make the correspondence explicit in terms of Langlands parameters.