Uniform decomposition of the flag scheme by a symmetric subgroup action

TL;DR

This paper develops a scheme-theoretic framework for decomposing flag varieties under symmetric subgroup actions, extending classical orbit classification results and proving affine embedding properties.

Contribution

It introduces a scheme-theoretic analog of orbit decompositions, establishing affineness of models and analyzing classical examples under new assumptions.

Findings

Scheme-theoretic orbit decompositions are established.

Models of orbits are affinely embedded into the flag scheme.

Classical examples illustrate scheme-theoretic consequences.

Abstract

In this paper, we establish a scheme-theoretic analog of the works of Matsuki and Richardson--Springer on the symmetric subgroup orbit decomposition of the flag variety under a certain assumption that asserts local constancy of their combinatorial description of the classification of orbits at geometric points (the local constancy hypothesis). We also prove that scheme-theoretic models of orbits are affinely imbedded into the flag scheme under the same assumption (Beilinson--Bernstein's affinity theorem). Finally, we compute some classical examples to give scheme-theoretic consequences of the geometric point free and descent phenomena of orbit decompositions.