Let $X$ be a smooth projective curve over $\mathbb{C}$ and Let $G$ be a complex reductive group. By a parahoric group scheme $\mathcal{G}$ over $X$, I mean a smooth group scheme over $X$ whose restriction to an open set $U$ is the constant group scheme $U \times G$ (or $\mathcal{G}_{K(X)} \cong spec(K(X)) \times G)$ and for $x \in X - U $, if $\mathcal{O}_{X,x}$ is the completion of the local ring at $x$ and $K_x$ its fraction field, then $\mathcal{G}(\mathcal{O}_{X,x}) \subset \mathcal{G}(K_x)$ is a parahoric sub-group in the sense of Bruhat-Tits. I am interested in the case when these local parahoric subgroups are just the Iwahori or the inverse image of a fixed borel subgroup $B \subset G$ under the evaluation maps ($G(\mathcal{O}_{X,x}) \rightarrow G(\mathbb{C})$). Is it true then that $$ B \subset \mathcal{G}(X)? $$ or if not is there a description of the groups $\mathcal{G}(X)$?.
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1$\begingroup$ For any Dedekind scheme $X$ and dense open $U \subset X$, an $X$-affine scheme $Y$ is "the same" as a $U$-affine scheme $Y'$ and $\widehat{O}_{X,x}$-schemes $Y'_x$ equipped with identifications with the generic fiber of $Y'$ over ${\rm{Frac}}(\widehat{O}_{X,x})$. This interacts well with smoothness, group scheme structure, etc. See Prop. D4(b) in section 6.2 of "Neron models". Hence, you can pre-assign whatever you want over each $\widehat{O}_{X,x}$. So using the "same" $B$ at each $x \in X-U$ does the job. This is why the real content of Bruhat-Tits theory is in the local case. $\endgroup$– user54268Commented Aug 16, 2014 at 17:37
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$\begingroup$ hi thank you , yes i am aware of the patching result you mentioned above and hence suspected the inclusion $B \subset \mathcal{G}(X)$. When all the parahorics under consideration are inside $G(\mathcal{O}_x)$,we have a map $\mathcal{G} \rightarrow C \times G$,so is it the case that $\mathcal{G}(X) \subset G$? $\endgroup$– rvarmaCommented Aug 16, 2014 at 18:14
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