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Let K/F be a quadratic extension of non archimedean local fields. Let V be anirreducible 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 isnot a supercuspidal representation. Let D be a division algebra of center F andindex 2r. In the following work, we try to answer this question : Is therepresentation V D^{*}-distinguished if and only if its image by theJacquet-Langlands correspondence is GL2(F)-distinguished. To answer thisquestion, we use the coefficients system on the Bruhat-Tits building of Gassociated to the representation V by P. Schneider and U. Stuhler and theparametrization of level zero discrete series representation given by J.Silberger and E. W. Zink.

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