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

TL;DR

This paper proves that the $A^1$-Euler characteristic of the variety of maximal tori in a reductive group over a field is invertible in the Grothendieck-Witt ring, confirming a conjecture by Morel and enabling a generalized splitting principle.

Contribution

It establishes the invertibility of the $A^1$-Euler characteristic in $ ext{GW}(k)$ for the variety of maximal tori, settling a conjecture and deriving a new splitting principle.

Findings

The $A^1$-Euler characteristic is invertible in $ ext{GW}(k)$.

The result confirms a conjecture by Fabien Morel.

A generalized splitting principle is derived from the main theorem.

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.