PARAMETRIZATION, STRUCTURE AND BRUHAT ORDER OF CERTAIN SPHERICAL QUOTIENTS

Let G be a reductive algebraic group and let Z be the stabilizer of a nilpotent element e of the Lie algebra of G. We consider the action of Z on the flag variety of G, and we focus on the case where this action has a finite number of orbits (i.e., Z is a spherical subgroup). This holds for instance if e has height 2. In this case we give a parametrization of the Z-orbits and we show that each Z-orbit has a structure of algebraic affine bundle. In particular, in type 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, we show that the Bruhat order of the Z-orbits can be described in this way.