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I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of it, and its applications concerning the Hecke Algebra. The subgroups play an important role in the work of M.F. Vigneras as evident from the appendix of this paper .

My background: I know the basic theory of connected reductive groups (eg, book of Humphreys) I know the basic elements of representation theory (eg, induction, restriction, parabolic induction and restriction, basic definition of relative Hecke algebra for locally profinite groups)

Thank you for your help.

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  • $\begingroup$ By the way, at least among my circle, the use "pro-$p$ Iwahori" is non-standard. (My first thought would be "but the Iwahori isn't pro-$p$ …".) I would say pro-unipotent radical or, if you want to think of abstract pro-finite groups, "Sylow pro-$p$-subgroup of the Iwahori." $\endgroup$
    – LSpice
    Commented May 28, 2020 at 19:00

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The definition is quite simple. Let $\mathcal O$ be the ring of integers of $F$, which I shall assume is a local field of residual characteristic $p$. The standard pro-$p$-Iwahori is the group of matrices in $GL_n(\mathcal O)$ that are upper unipotent modulo the maximal ideal $m$ of $\mathcal O$. Any conjugate of that group is called a pro-$p$-Iwahori.

Now I note that in your background, you don't mention Iwahori. If you are not familiar with classical, not pro-$p$, Iwahori subgroups (same definition as above but with unipotent replaced by triangular), I think you should lear this things first. That will lead you to Bruhat-Tits building, which anyways play a role in Vignéras work, even if not always advertised. References for this are the original article of Bruhat-Tits in IHES, the paper of Tits in Corvallis (that is PSPM 33), and many books that have been written on buildings.

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    $\begingroup$ Even the original paper by Iwahori-Matsumoto is useful for this particular example: ams.org/mathscinet-getitem?mr=0185016 (the paper itself is freely available online through numdam.org, as are the papers by Bruhat-Tits, all in old IHES volumes). However, most of the early literature is in French. $\endgroup$
    – Jim Humphreys
    Commented Oct 5, 2015 at 21:33

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