Parametrization, structure and Bruhat order of certain spherical quotients

Abstract

Let G G be a reductive algebraic group and let Z Z be the stabilizer of a nilpotent element e e of the Lie algebra of G G . We consider the action of Z Z on the flag variety of G G , and we focus on the case where this action has a finite number of orbits (i.e., Z Z is a spherical subgroup). This holds for instance if e e has height 2 2 . In this case we give a parametrization of the Z Z -orbits and we show that each Z Z -orbit has a structure of algebraic affine bundle. In particular, in type A A , we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type A A , we show that the Bruhat order of the Z Z -orbits can be described in this way.