On the 𝔸1-Euler Characteristic of the Variety of Maximal Tori in a Reductive Group

Abstract

Abstract 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.