Parametrization, structure and Bruhat order of certain spherical quotients

Chaput, Pierre-Emmanuel; Fresse, Lucas; Gobet, Thomas

doi:10.1090/ert/584

Parametrization, structure and Bruhat order of certain spherical quotients
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by Pierre-Emmanuel Chaput, Lucas Fresse and Thomas Gobet

Represent. Theory 25 (2021), 935-974

DOI: https://doi.org/10.1090/ert/584

Published electronically: October 21, 2021

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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.

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