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Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of anilpotent 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 afinite number of orbits (i.e., $Z$ is a spherical subgroup). This holds forinstance 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 affinebundle. In particular, in type $A$, we deduce that each orbit has a naturalcell decomposition. In the aim to study the (strong) Bruhat order of theorbits, we define an abstract partial order on certain quotients associated toa Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbitscan be described in this way.

Parametrization, structure and Bruhat order of certain spherical quotients