Oriented embedding functors of tori as homogeneous spaces

TL;DR

This paper characterizes certain homogeneous spaces under reductive group schemes with maximal tori as stabilizers, addressing a specific case over semilocal bases and applying results to local-global principles for algebra embeddings.

Contribution

It provides a new characterization of homogeneous spaces with maximal tori as stabilizers, especially in the quasi-split case, and applies this to algebra embedding problems.

Findings

Characterization of homogeneous spaces with maximal tori as stabilizers

Answer to Levine's question on homogeneous SL$_n$-spaces in the quasi-split case

Application to local-global principles for algebra embeddings

Abstract

We provide a characterization of homogeneous spaces under a reductive group scheme such that the geometric stabilizers are maximal tori. The quasi-split case over a semilocal base is of special interest and permits to answer a question raised by Marc Levine on homogeneous SL$_n$-spaces. At the end, we provide an application to the local-global principles for embeddings of \'etale algebras with involution into central simple algebras with involution.