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Variétés de drapeaux symplectiques impaires

Abstract : Symplectic grassmannians and, more generally, symplectic flag manifolds, are the varieties of isotropic subspaces, respectively flags of isotropic subspaces, with respect to a nondegenerate skew form. These are the projective homogeneous varieties of the symplectic group.
We study odd symplectic grassmannians and flag manifolds, which are analogous objects defined with respect to a generic skew form on an odd dimensional complex vector space. These varieties are provided with natural actions of the odd symplectic group of linear transformations preserving the skew form. We show that even if these actions are not transitive, these varieties share a lot of properties with homogeneous varieties.
In particular, we compute the automorphisms groups of odd symplectic grassmannians and obtain that all these automorphisms come from the action of the odd symplectic group. We establish a Borel-Weil type theorem for the odd symplectic group and explain the relation between certain classes of representations of this group constructed by Proctor and by Shtepin.We study as well the equivariant cohomology of the variety of maximal odd symplectic flags. We obtain a Chevalley-Pieri type formula and we describe a Borel presentation of the equivariant cohomology ring. From the latter we infer that the ordinary cohomology ring of the variety of maximal odd symplectic flags is isomorphic with the ordinary cohomology ring of the variety of quadratic flags.
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Contributor : Martine Barbelenet <>
Submitted on : Thursday, December 8, 2005 - 12:13:53 PM
Last modification on : Wednesday, November 4, 2020 - 2:17:56 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:34:26 PM


  • HAL Id : tel-00011170, version 1



Ion Alexandru Mihai. Variétés de drapeaux symplectiques impaires. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2005. Français. ⟨tel-00011170⟩



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