Parahoric Group Scheme

Question

I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!

thanks

Answer 1

Well, I won't give a definition, but I can make some comments which may (?) be somewhat helpful.

The parahoric group schemes associated to a reductive group $G$ over a local field $F$ are certain smooth, affine group schemes $\mathcal{P}$ over the ring of integers $R$ of $F$ with the property that their generic fibers $\mathcal{P}_F$ identify with the given reductive group $G$.

Up to "connectedness phenomena" (which I'm going to ignore here) the group of $R$-points $\mathcal{P}(R)$ of a parhoric group scheme is the stabilizer in $G(F)$ of a point $x$ in the "Bruhat-Tits building" (a certain geometric gadget on which $G(F)$ acts). Sometimes one might call this group of points $\mathcal{P}(R)$ a "parahoric subgroup". Note that if $R$ has finite residue field, then $\mathcal{P}(R)$ is (or identifies with) a compact open subgroup of the locally compact group $G(F)$.

Choice of a parahoric group scheme $\mathcal{P}$ involves in part the choice of a maximal $F$-split torus $T$ of $G$. Then there is an $R$-split torus $\mathcal{T}$ contained in $\mathcal{P}$, and under the identification of $G$ with $\mathcal{P}_F$, the torus $T$ identifies with $\mathcal{T}_F.$ At least when $G$ is split, the point $x$ mentioned above determines for each root $\alpha$ a certain $R$-form of the root group $U_\alpha$, and the parahoric $\mathcal{P}$ is generated by $\mathcal{T}$ together with these $R$-forms of the root subgroups.

In general, $\mathcal{P}$ is not a reductive group scheme.

I hope the above suggests that the parahoric group schemes have interesting and important properties, but not really an "easy to write down definition". The "survey" paper of Jacque Tits [*] in the Corvallis proceedings is perhaps a good place to start learning about them.

[*] Reductive groups over local fields. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 29–69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

Answer 2

What George McNinch has written is most helpful, but maybe I can point out an obstacle to locating the sort of precise reference you ask for. The most relevant work of Bruhat-Tits, apart from surveys such as the one by Tits in English which George refers to, is found online at numdam.org, starting with their IHES papers I and II. In each paper there are many separate occurrences of the terms schema and sous-groupes parahoriques but apparently no explicit formulation of the definition you want. Another paper by them here may also be useful. Their style however is quite formal, as you realize. (I guess it's clear that the artificial term parahorique combines the notions of parabolic subgroup and Iwahori subgroup in the case of local fields.)