  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.
