Abstract
The universal centralizer of a semisimple algebraic group G is the family of centralizers of regular elements, parametrized by their conjugacy classes. When G is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification \(\overline{{\mathcal {Z}}}\) of the universal centralizer \({\mathcal {Z}}\) by taking the closure of each fiber in the wonderful compactification \({\overline{G}}\). We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of \(\overline{{\mathcal {Z}}}\). We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties—we apply this connection to compute the singular cohomology of \(\overline{{\mathcal {Z}}}\), and to study the geometry of the corresponding universal Hessenberg family.
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Acknowledgements
The author would like to thank Victor Ginzburg, Sam Evens, Ioan Mărcuţ, Sergei Sagatov, and Travis Schedler for many interesting discussions. Part of this work was completed while the author was supported by a National Science Foundation MSPRF under award DMS–1902921.
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