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



Parahoric Group Scheme
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](http://www.ams.org/mathscinet-getitem?mr=546588) [\*] 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.


