**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.



Naive definition of parahoric subgroup
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](http://www.ams.org/books/pspum/033.1) §3.5.3, or Proposition 4.6.32 of [BT2](http://www.numdam.org/item?id=PMIHES_1984__60__5_0), 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](http://www.ams.org/books/pspum/033.1)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=546588)), and Théorème 4.6.33 of [BT2](http://www.numdam.org/item?id=PMIHES_1984__60__5_0) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=756316)). 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](https://books.google.com/books?id=7oBZIxZfPQAC&pg=PA53)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=3525846)) in the Ottawa proceedings, or his paper "[Smooth models associated to concave functions in Bruhat–Tits theory](http://www2.ims.nus.edu.sg/preprints/2002-20.pdf)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=3525846); 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.


