B_{n-1}-bundles on the flag variety, I

TL;DR

This paper establishes a recursive method to classify B_{n-1}-orbits on flag varieties for GL(n) and SO(n), revealing their bundle structure and paving the way for explicit combinatorial classifications.

Contribution

It introduces a uniform inductive approach to classify B_{n-1}-orbits on flag varieties for GL(n) and SO(n), extending previous work and enabling explicit orbit descriptions.

Findings

Orbits form bundles over smaller flag varieties.

Develops an inductive classification procedure.

Lays groundwork for explicit combinatorial orbit classification.

Abstract

We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify $B_{n-1}$-orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and determine completely the closure relation between orbit closures. This further develops work of Hashimoto in the general linear group case.