What is the Iwahori subgroup for $PGL\_2(F)$ where $F$ is a local field? I am also looking for the Levi subgroups but it seems that there is only 1 levi subgroup been the identity but this seems odd to me. 



Iwahori for PGL_2 
The (standard) Iwahori $\bar I$ of ${\rm PGL}(N,F)$ is the image of the Iwahori subgroup $I$ of ${\rm GL}(N,F)$ formed of the matrices with integer coefficients, upper triangular mod ${\mathfrak p}\_F$, and with invertible determinant. To extract its defition from Bruhat-Tits general theory, there is an extra difficulty here coming from the fact that the group is not simply connected (contrary to e.g. ${\rm SL}(N,F)$). 


