We show that for a reductive group $G$ over a field $k$ the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck–Witt ring ${\textrm{GW}}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle that allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.

中文翻译：

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还原群中最大环面多样性的𝔸1-Euler特征

我们证明，对于域 $k$ 上的约简群 $G$，$G$ 中最大环面多样性的 $\mathbb{A}^1$-Euler 特征是 Grothendieck-Witt 环 $ 的可逆元素{\textrm{GW}}(k)$，解决了 Fabien Morel 猜想的弱形式。作为一个应用，我们获得了一个广义分裂原理，它允许将 Nisnevich 局部平凡 $G$-torsor 的结构组减少为最大环面的归一化器。