# Adhérences d'orbites des sous-groupes de Borel dans les espaces symétriques

Abstract : This thesis deals with singularities of orbits closures of Borel subgroups in symmetric spaces. We consider a reductive group $G$, and the fixed points subgroup $H$ of an involution of $G$. Following Richardson and Springer we parameterize the orbits of a Borel subgroup in the symmetric space $G/H$, and we give a combinatorial description of those orbits closures. We also construct slices to describe singularities of those orbits closures. We study more specifically the symmetric space $PSL_n/PSO_n$. In this case, thanks to the combinatorial description and to the slices, we give some orbits closures normality criteria and a characterization of smoothness in codimension one. Finally, we give many examples of orbits closures of a Borel subgroup in symmetric spaces with different kinds of singularities : non normal orbits closures of codimension one in $G/H$, and orbits closures that are neither normal nor Cohen-Macaulay.
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https://tel.archives-ouvertes.fr/tel-00000888
Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, November 26, 2001 - 9:06:36 AM
Last modification on : Wednesday, November 4, 2020 - 1:57:18 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 1:10:10 PM

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• HAL Id : tel-00000888, version 1

### Citation

Stéphane Pin. Adhérences d'orbites des sous-groupes de Borel dans les espaces symétriques. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2001. Français. ⟨tel-00000888⟩

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