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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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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.
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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