Let F be a p-adic field. In this talk I'll study the Om(F)-distinction of some specific principal series representations of Glm(F). The main goal is to give a computing method to see if those representations are distinguished or not so we can also explicitly find a non zero Om(F)-equivariant linear form. This linear form will be given by the integral of the representation's matrix coefficient over Om(F).
After explaining on what specific principal series representations I'm working and why I need those specificities, I'll explain the different steps to compute the integral of my representation's matrix coefficient over Om(F). I'll explicitly give the obtained result for the case m=3. After that I'll explain an asymptotic result we can obtain when we can't compute the integral explicitly.