Correspondance de Jacquet-Langlands et distinction : cas de certaines series discretes non cuspidales de niveau 0

TL;DR

This paper investigates the relationship between Jacquet-Langlands correspondence and distinction properties of certain non-cuspidal level zero discrete series representations of GL2 over division algebras, using geometric and parametrization tools.

Contribution

It establishes criteria for when a representation is distinguished under D^{*} versus its Jacquet-Langlands image, linking distinction properties through geometric and algebraic methods.

Findings

V is D^{*}-distinguished iff its Jacquet-Langlands image is GL2(F)-distinguished.

Uses Bruhat-Tits building coefficients system for analysis.

Provides new insights into distinction criteria for non-cuspidal series.

Abstract

Let K/F be a quadratic extension of non archimedean local fields. Let V be an irreducible level zero discrete series representation of the group G = GL2(R), where R is a division algebra of center K and of index r. One assumes that V is not a supercuspidal representation. Let D be a division algebra of center F and index 2r. In the following work, we try to answer this question : Is the representation V D^{*}-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL2(F)-distinguished. To answer this question, we use the coefficients system on the Bruhat-Tits building of G associated to the representation V by P. Schneider and U. Stuhler and the parametrization of level zero discrete series representation given by J. Silberger and E. W. Zink.