On Bruhat-Tits theory over a higher dimensional base

TL;DR

This paper generalizes Bruhat-Tits theory to higher-dimensional bases, constructing smooth group schemes associated with concave functions on root systems, extending applications to group schemes on embeddings and surface singularities.

Contribution

It introduces higher-dimensional Bruhat-Tits groups as schematic, smooth group schemes compatible with normal crossing divisors, extending classical theory to multiple variables.

Findings

Defined n-bounded subgroups as generalizations of Bruhat-Tits groups

Proved these groups are valued points of smooth quasi-affine group schemes

Extended results to rings with additional variables and applications in characteristic zero

Abstract

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}.