Oriented embedding functors of tori as homogeneous spaces

TL;DR

The paper addresses the problem of classifying homogeneous spaces X with reductive G such that geometric stabilizers are maximal tori, proving that X is the oriented embedding space E(G, Psi, v) for some admissible Psi and orientation v. The approach centers on twisted root data and the orientation framework Isomext/Isomint, introducing embedding functors Emb(G, Psi) and their oriented versions, and showing that X is isomorphic to an Emb(G, Psi, v) under mild hypotheses; it further develops Weil restriction and isotypic decompositions to reduce complex cases to simpler components. The LG-ring generalization extends the theory to base schemes with LG properties, using algebraic spaces to construct (4)-versal torsors and establishing a local-global principle for embeddings of étale algebras with involution into central simple algebras with involution when the unitary group is quasi-split. The results yield practical criteria for embedding problems, provide a decomposition framework via isotypic components, and culminate in an arithmetic application to local-global principles that connect root data, torus embeddings, and automorphism structures in both classical and LG settings.

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 étale algebras with involution into central simple algebras with involution.

Theorems & Definitions (52)

• Remark 2.1
• Lemma 2.2
• proof
• Example 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.2.1
• proof
• Proposition 3.2.2