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The universal centralizer of a semisimple algebraic group is the family of centralizers of regular elements, parametrized by their conjugacy classes. When the group is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification of the universal centralizer by taking the closure of each fiber in the wonderful compactification. We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of this new space. We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties—we apply this connection to compute the singular cohomology of the compactification, and to study the geometry of the corresponding universal Hessenberg family.
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Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties
We use the Springer correspondence to give a partial characterization of the irre- ducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kahler package.”
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- Award ID(s):
- 1902921
- PAR ID:
- 10479050
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Transformation Groups
- ISSN:
- 1083-4362
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00031-022-09755-3
