Bruhat-Tits theory over a smooth higher dimensional base

Abstract: We report on a joint work with Vikraman Balaji. It addresses the
following question:

Let $G$ be an almost simple, split, simply connected Chevalley group
scheme over $\mathbb{Z}$. Let $\mathbb{A}^\circ$ denote the complement of
the "axes" in $\mathbb{A}^n_{_k}$, and let $U$ be its union with $Spec$ of
DVRs which are the local rings at the generic points of the axes in
$\mathbb{A}^n_{_k}$. Given BT group schemes adapted to the axis divisors
of $\mathbb{A}^n_{_k}$, using the identity function, we glue them with $G
\times \mathbb{A}^\circ$ to get a group scheme on $U$. Does it extend to
the whole space $\mathbb{A}^n_{_k}$?

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