Building ofGL(m, D) and centralizers

LetG=GL(m, D) whereD is a central division algebra over a commutative nonarchimedean local fieldF. LetE/F be a field extension contained inM(m, D). We denote byI (resp.I_E) the nonextended affine building ofG (resp. of the centralizer ofE^x inG). In this paper we prove that there exists a uniqueG_E-equivariant affine mapj_EI_EI. It is injective and its image coincides with the set ofE^x-fixed points inI. Moreover, we prove thatj_E is compatible with the Moy-Prasad filtrations.This author's contribution was written while he was a post-doctoral student at King's College London and supported by an european TMR grant