On Bruhat-Tits theory over a higher dimensional base

Let \(k\) be a perfect field. Assume that the characteristic of \(k\) satisfies certain tameness assumptions \eqref{tameness}. Let \(\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket\) and set \(K_{_n} := \text{Fract}~\cO_{_n}\). Let \(G\) be an almost-simple, simply-connected affine Chevalley group scheme with a maximal torus \(T\) and a Borel subgroup \(B\). Given a \(n\)-tuple \({\bf f} = (f_{_1}, \ldots, f_{_n})\) of concave functions on the root system of \(G\) as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we define {\it {\tt n}-bounded subgroups \({\tt P}_{_{\bf f}}\subset G(K_{_n})\)} as a direct generalization of Bruhat-Tits groups for the case \(n=1\). We show that these groups are {\it schematic}, i.e. they are valued points of smooth {\em quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it adapted to the divisor with normal crossing \(z_1 \cdots z_n =0\)} in the sense that the restriction to the generic point of the divisor \(z_i=0\) is given by \(f_i\) (resp. sums of concave functions given by points of the apartment). This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In \S\ref{mixedstuff}, under suitable assumptions on \(k\) \S \ref{charassum}, we extend all these results for a \(n+1\)-tuple \({\bf f} = (f_{_0}, \ldots, f_{_n})\) of concave functions on the root system of \(G\) replacing \(\mathcal O_{_n}\) by \({\cO} \llbracket x_{_1},\cdots,x_{_n} \rrbracket\) where \(\cO\) is a complete discrete valuation ring with a perfect residue field \(k\) of characteristic \(p\). In the last part of the paper, we give applications in char zero to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of {\tt 2-parahoric} group schemes on minimal resolutions of surface singularities that arose in \cite{balaproc}.