On Bruhat-Tits theory over a higher dimensional base

Balaji, Vikraman ; Pandey, Yashonidhi

Épijournal de Géométrie Algébrique, Tome 8 (2024)

Résumé

Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies certain tameness assumptions (). 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 , , we define 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 schematic, i.e. they are valued points of smooth quasi-affine (resp. affine) group schemes with connected fibres and 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 , under suitable assumptions on $k$ , 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 2-parahoric group schemes on minimal resolutions of surface singularities that arose in .

Détail

DOI : 10.46298/epiga.2024.9759

Classification : 14D20, 14L15, 14M27