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.
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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 algebraic-groups would be more appropriate here than rt. $\endgroup$– Jim HumphreysSep 16, 2011 at 14:37
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$\begingroup$ 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 p-adic 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 MoraSep 17, 2011 at 2:52
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$\begingroup$ 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 BroussousSep 17, 2011 at 14:31
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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 non-reductive linear algebraic group over the residue field (which might have a Levi factor). $\endgroup$– George McNinchSep 21, 2011 at 16:00
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$\begingroup$ By the way, your remark about the number of Levi subgroups is basically correct, but your identification is not: there is only one proper Levi subgroup, up to rational conjugacy, and it is the split torus. $\endgroup$– LSpiceJun 30, 2020 at 7:53
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1 Answer
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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. ${\rm SL}(N,F)$).
answered Sep 16, 2011 at 7:11
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$\begingroup$ Paul Broussous do you know of any paper that has types and covers for PGL_2. Or GL_2 $\endgroup$ Sep 18, 2011 at 21:09
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$\begingroup$ 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. $\endgroup$ Sep 19, 2011 at 7:48