# Naive definition of parahoric subgroup

Background

Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, encapsulated in the theory of Bruhat-Tits. There they associate to $G$ a combinatorial object called a building $B(G,F)$ upon which $G(F)$-actions, and for which many classes of 'nice' compact open subgroups of $G(F)$ can be understood as stabilizers of certain 'nice' combinatorial subobjects of $B(G,F)$. These are incredibly valuable since these 'nice' compact open subgroups of $G(F)$ are pivotal in understanding many aspects of the structure of $G(F)$ (e.g. in the construction of representations of $G(F)$ in the form of the Moy-Prasad filtration or as 'reasonable levels' in Shimura varieties).

This combinatorial data can be somewhat daunting, and so it is sometimes (at least for me) more convenient to have a way of understanding these 'nice' compact open subgroups of $G(F)$ without needing to understand the inner workings of the Bruhat-Tits machine.

As an example, one type of 'nice' compact open subgroup of $G(F)$ are the so-called hyperspecial subgroups. One can define these purely in terms of $B(G,F)$, but one can also define them as $\mathcal{G}(\mathcal{O}_F)$ where $\mathcal{G}$ is a reductive model of $G$ over $\mathcal{O}_F$. This allows one to work with hyperspecial subgroups without having to understand the whole definition of $G(\mathcal{O}_F)$.

The question below is whether another class of 'nice' subgroups can be understand in a similar way.

Actual question

Let $F$ be a $p$-adic local field with ring of integers $\mathcal{O}_F$ and residue field $k$. Let $G$ be a connected unramified reductive group over $F$. To what extent is it true that parahoric subgroups of $G(F)$ are pullbacks of parabolic subgroups of $\mathcal{G}_k$ for $\mathcal{G}$ a reductive model of $G$ over $\mathcal{O}_F$?

Namely, if $\mathcal{G}$ is a reductive model of $G$ over $\mathcal{O}_F$, and $P$ is a parabolic subgroup of $\mathcal{G}_k$, then is $\mathrm{red}^{-1}(P(k))\subseteq \mathcal{G}(\mathcal{O}_F)$ a parahoric subgroup of $G(F)$ (here $\mathrm{red}:\mathcal{G}(\mathcal{O})\to \mathcal{G}(k)$ is the reduction map) a parahoric subgroup of $G(F)$? Do they all arise in this manner?

I know that one can define parahorics as the $\mathcal{O}_F$-points of 'parahoric group schemes', but the definition of these is very opaque to me. Do these generalize the above construction in some sense?

If one can say more, feel free to assume that $G$ is split.

• This is an answer to your comment below, but I put it here because it's always worth advertising as broadly as possible. Joe Rabinoff's senior thesis is an excellent way of 'grokking' parahorics, and MP subgroups more generally. It achieves this at some expense of generality, but, in my experience, when you need the more general results you'll have enough experience to look it up. Apr 23 '18 at 17:01
• Oh, one more thing! You can very often get by without understanding the full building by appealing to the fact that any two facets in the building lie in a common apartment, and that apartments are affine spaces under the cocharacter lattice of a maximal split torus (tensored with $\mathbb R$). In practice, this simplifies a lot of computations. Although I wouldn't necessarily recommend it for learning the theory, Schneider–Stuhler §1.1 (MSN) takes this approach. Apr 24 '18 at 19:01

I'm not sure what it means to define parahoric subgroups "purely in terms of $B(G, F)$"; I would say that every definition boils down to taking integral points of integral models in one way or another. By the way, it is not true in general that parahoric subgroups are full facet stabilisers; in general, the group scheme $\mathcal G$ underlying the full stabiliser is disconnected, and one must pass to its identity component before taking integral points in order to get the parahoric. (See nonetheless, say, Tits §3.5.3, or Proposition 4.6.32 of BT2, where it is observed that one does have equality for simply connected groups.)

Nonetheless, the result you want (that parahoric subgroups are pullbacks of parabolic subgroups of parahoric subgroups) is correct; it is §3.5.4 of Tits's Corvallis article "Reductive groups over local fields" (MSN), and Théorème 4.6.33 of BT2 (MSN). Of course, it doesn't really 'reduce' the problem, since one still needs to have the original parahoric subgroup to pull back its parabolic subgroups. You may also find it helpful to read Yu's various expository articles (say, "Bruhat–Tits theory and buildings" (MSN) in the Ottawa proceedings, or his paper "Smooth models associated to concave functions in Bruhat–Tits theory" (MSN; I have linked to the preprint at NUS, which I have not compared to the published version)) for a modern perspective on BT theory; he had a program for a while to make their work more accessible. He used to have some notes available on his Purdue web page, but that no longer exists, and he doesn't seem to have migrated them to CUHK.

• Thanks for the speedy answer! I am not sure I entirely follow though. Secdtion 3.5.4 of Tit's Corvallis article doesn't seem helpful, although it's a little hard to decipher exactly what it's talking about. I also find 4.6.32 of BTII a little difficult to parse. I'll take a look at Yu's article. Let me make a clarificatory remark about my question that maybe was not so clear. What I want to know is the following: can you understand parahorics in any way that doesn't require fixing extra data. For example, to define hyperspecial you don't need to fix a pinning, or anything of the sort Apr 23 '18 at 10:09
• is that possible for parahorics? Can someone understand them without really understanding BT theory. I ask because, as one might surmise, I don't really understand BT theory, but would like to understand parahorics, at least intuitively, say for split or unramified groups. Thanks very much! Apr 23 '18 at 10:10
• Anent §3.5.4, note that the spherical building is a way of parameterising the parabolic subgroups, and the link of a facet $F$ in the building is—well, I don't have the formal definition to hand, but it's essentially the collection of facets whose closures contain $F$. Thus, the claim there is that there is a canonical bijection between the parabolic subgroups of $(\mathcal G_F)_k$ and that collection of facets. In particular, every parahoric subgroup arises by pulling back a parabolic subgroup of a parahoric at a vertex (but, sadly, not necessarily a special one). Apr 23 '18 at 17:05
• I guess what I mean to say is: what you want to be true is true, so, if it was just a binary question, then you're done and (for this purpose) don't need to get into the weeds of BT theory. I provide the reference not necessarily for clarification (though Tits's Corvallis article is a great introduction), but rather to give you something to cite if you need it. Apr 23 '18 at 17:51
• Thanks! So, for $G$ unramified, the parahorics are exactly the pullbacks of parabolics of models of $G$--or is this only true for split $G$? Thanks again. :) Apr 24 '18 at 5:15

For another perspective on parahoric subgroups, I find the appendix by Haines-Rapoport (here) quite useful. They show that the parahoric subgroup attached to a facet $\mathcal{F}$ is simply the intersection of the pointwise stabilizer of $\mathcal{F}$ in $G(F)$ with the kernel of the Kottwitz homomorphism (you may have to take Frobenius fixed points somewhere). In particular, if your group $G$ is semisimple and simply connected, then the Kottwitz homomorphism is trivial, and parahoric subgroups are just pointwise stabilizers of facets.

• At least from my point of view, this is the most useful perspective for computations. I worked with parahorics for a while before I learned about this appendix, and I remember that it dramatically changed how I thought about them. Apr 23 '18 at 20:19
• Thanks! I did see this article, and I'm sure it gives some zen understanding of pararhorics, but it still requires an understanding of some (perhaps basic) BT theory, which I am trying to avoid. I will keep it in mind for later usage though! Apr 24 '18 at 5:16

A poster on parahoric subgroup http://www-personal.umich.edu/~fintzen/Jessica_Fintzen_poster_stable_vectors.pdf

Video explanation: https://youtu.be/UO_RZzaBTmc

• Jessica's poster is great, but it doesn't seem to say anything about the specific question, and not much more about parahoric subgroups than a few specific examples. Mar 31 '20 at 19:37