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Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, what is the precise reference/reason for this fact?

I strongly suspect that the answer is yes--my understanding is that these are the "boring" examples of parahorics--but the Bruhat-Tits papers are difficult to navigate and the answer seems to explicitly appear neither in BT I (which is abstract group theory anyway in a some sense, it seems) nor in BT II. I realize that there are two possible meanings for "parahoric": as a group scheme model of $G_K$ or as a subgroup $G(R) \subset G(K)$ (the latter is defined in BT II through BT buildings). An answer addressing either one of these senses would be most welcome; I am interested in knowing both.

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    $\begingroup$ When the residue field is perfect, the answer is yes, and they are called hyperspecial parahorics. I'd recommend Tits' Corvallis article as a reference. The point is that Bruhat-Tits and parahorics are meant to behave well under unramified base change, after which your group (scheme) becomes split. (My Bruhat-Tits theory works only for complete DVR with perfect residue field - most things should still work for Henselian, but I am not capable of carefully telling. Nor I am capable of saying what happen with non-perfect residue fields.) $\endgroup$
    – Cheng-Chiang Tsai
    Commented Oct 12, 2018 at 20:42

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