Bn−1-bundles on the flag variety, I

Abstract

In this paper, we consider the orbits of a Borel subgroup Bn−1 of Gn−1=GL(n−1) (respectively SO(n−1)) acting on the flag variety Bn of G=GL(n) (resp. SO(n)). The group Bn−1 acts on Bn with finitely many orbits, and we use the known description of Gn−1-orbits on B to study these orbits. In particular, we show that a Bn−1-orbit is a fibre bundle over a Bn−1-orbit in a generalized flag variety of Gn−1 with fibre a Bm−1-orbit on Bm for some m<n. We further use this fibre bundle structure to study the orbits inductively and describe the monoid action on the fibre bundle. In a sequel to this paper, we use these results to give a complete combinatorial model for these orbits and show how to understand the closure relations on these orbits in terms of the monoid action.