Parametrization, structure and Bruhat order of certain spherical quotients

Let GG be a reductive algebraic group and let ZZ be the stabilizer of a nilpotent element ee of the Lie algebra of GG. We consider the action of ZZ on the flag variety of GG, and we focus on the case where this action has a finite number of orbits (i.e., ZZ is a spherical subgroup). This holds for i...

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Bibliographic Details
Published in:Representation theory 2021-10, Vol.25 (33), p.935-974
Main Authors: Chaput, Pierre-Emmanuel, Fresse, Lucas, Gobet, Thomas
Format: Article
Language:English
Subjects:
Algebraic Geometry
Combinatorics
Mathematics
Representation Theory
Research article
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Summary:Let GG be a reductive algebraic group and let ZZ be the stabilizer of a nilpotent element ee of the Lie algebra of GG. We consider the action of ZZ on the flag variety of GG, and we focus on the case where this action has a finite number of orbits (i.e., ZZ is a spherical subgroup). This holds for instance if ee has height 22. In this case we give a parametrization of the ZZ-orbits and we show that each ZZ-orbit has a structure of algebraic affine bundle. In particular, in type AA, 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 AA, we show that the Bruhat order of the ZZ-orbits can be described in this way.
ISSN:1088-4165
1088-4165
DOI:10.1090/ert/584