Parametrization, structure and Bruhat order of certain spherical quotients

Abstract

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.a partial order on the quotient W/W θ L , where W θ L stands for the subgroup of fixed points of θ. We mostly address the situation where W θ L is a diagonal subgroup of W L . We investigate certain properties of this order (minimal representatives, cover relations).In type A n−1 , for a nilpotent element e of height 2, the Z G (e)-orbits of the flag variety B are parametrized by a quotient of the above-mentioned form, namely S n /(∆S r × S n−2r ), where ∆S r stands for the diagonal embedding of S r into S r × S r . Then, translating the results of [1,5] into our framework, we show that our combinatorial order coincides with the Bruhat order of the Z G (e)-orbits.