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.

1$\begingroup$ It's not clear what kind of source material you are starting with or what motivates the question, which seems to be a matter of standard terminology. (Some of the original papers like the 1965 IHES paper by Iwahori and Matsumoto can be freely accessed at www.numdam.org) Anyway, there are typically many Iwahori (resp. Levi) subgroups, together with suitable conjugacy theorems. The last part of the question is especially unclear to me, since the group considered is simple. And the tag algebraicgroups would be more appropriate here than rt. $\endgroup$ – Jim Humphreys Sep 16 '11 at 14:37

$\begingroup$ Are you Humphreys the one who wrote the book? If you are I am faltered that you comment on my question.The motivation comes from Bernstein theory for reductive padic groups.The bernstein components are index by conjugacy classes of levi subgroups and cuspidal representations on the levi...The Bernstein component is a subcategory and can be described as a category of modules over ring. This rings correspond to types and types come from open compact subgroups and representations on them.The Iwahori subgroup with trivial rep is an example of such a type that is the reason why the tag is rt. $\endgroup$ – Carlos De la Mora Sep 17 '11 at 2:52

$\begingroup$ In any reductive group the trivial character of "the" Iwahori subgroup (it is unique up to conjugacy) is a type for the Bernstein component corresponding to the minimal Levi subgroup together with its trivial representation. For ${rm SL}(2)$, that minimal Levi subgroup is the diagonal split torus. $\endgroup$ – Paul Broussous Sep 17 '11 at 14:31

1$\begingroup$ @Paul, By "Levi subgroup of G" you seem to mean "Levi factor of a parabolic subgroup of G". (I think I'm echoing part of Jim's comment about the original post). It is maybe worth being careful with the language in this context e.g. because the "special fiber" (="reduction mod p") of an Iwahori is a nonreductive linear algebraic group over the residue field (which might have a Levi factor). $\endgroup$ – George McNinch Sep 21 '11 at 16:00
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 BruhatTits general theory, there is an extra difficulty here coming from the fact that the group is not simply connected (contrary to e.g. ${\rm SL}(N,F)$).

$\begingroup$ Paul Broussous do you know of any paper that has types and covers for PGL_2. Or GL_2 $\endgroup$ – Carlos De la Mora Sep 18 '11 at 21:09

$\begingroup$ Both groups are considered in Roche's paper on types for principal series representations. Indeed the general setting of this paper is that of split groups. You can easily extract what you want. $\endgroup$ – Paul Broussous Sep 19 '11 at 7:48