## Parametrization, structure and Bruhat order of certain spherical quotients

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- by Pierre-Emmanuel Chaput, Lucas Fresse and Thomas Gobet PDF
- Represent. Theory
**25**(2021), 935-974 Request permission

## 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.## References

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## Additional Information

**Pierre-Emmanuel Chaput**- Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy F-54506, France
- MR Author ID: 697366
- Email: pierre-emmanuel.chaput@univ-lorraine.fr
**Lucas Fresse**- Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy F-54506, France
- MR Author ID: 875745
- Email: lucas.fresse@univ-lorraine.fr
**Thomas Gobet**- Affiliation: Institut Denis Poisson, CNRS UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France
- MR Author ID: 1076592
- Email: thomas.gobet@lmpt.univ-tours.fr
- Received by editor(s): September 18, 2020
- Received by editor(s) in revised form: June 2, 2021
- Published electronically: October 21, 2021
- Additional Notes: The first two authors were supported in part by the ANR project GeoLie ANR-15-CE40-0012. The third author was partially supported by the same project and by the ARC project DP170101579.
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 935-974 - MSC (2020): Primary 20G05, 17B08
- DOI: https://doi.org/10.1090/ert/584
- MathSciNet review: 4329194