On the ${\mathbb{A}}^1$-Euler characteristic of the variety of maximal tori in a reductive group

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 $\mathrm{GW}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle which allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.

13 pages, v2: minor revision

References in corpus (1)

• Additivity and Double Coset formulae for the Motivic and Étale Becker-Gottlieb transfer

Cited by in corpus (1)

• The Motivic Segal-Becker Theorem