Correspondance de Jacquet-Langlands et distinction : cas des representations cuspidales de niveau 0

TL;DR

This paper investigates the conditions under which level zero cuspidal representations of certain inner forms of general linear groups over quadratic extensions are distinguished, establishing a link via the Jacquet-Langlands correspondence.

Contribution

It determines the GL_m(D)-distinction criteria for level zero cuspidal representations of GL_p(R) and proves the equivalence of distinction under Jacquet-Langlands correspondence.

Findings

GL_m(D)-distinction conditions for level zero cuspidal representations

Equivalence of distinction between representations and their Jacquet-Langlands images

Characterization of distinguished representations over quadratic extensions

Abstract

Let K/F be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GL_m (D) be an inner form of GL_n (F) and GLp(R) = (M_m (D) \otimes K)^{\times} . Then GLp(R) is an inner form of GL_n (K). In this work, we determine conditions of GL_m (D)-distinction for level zero cuspidal representations of GL_p (R) which are the image of a level zero cuspidal representation by the Jacquet-Langlands correspondence, and we also prove that a level zero cuspidal representation of GL_n (K) is GL_n (F) distinguished if and only if its image by the Jacquet-Langlands correspondance is GL_m (D)-distinguished.