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The partial compactication of the universal centralizer

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The partial compactication of the universal centralizer
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19
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Let G be a semi simple algebraic group of adjoint type. The universa lcentralizer is the family of centralizers in G of regular elements in Lie (G), parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle T∗G. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of G. The symplectic structure extends to a log-symplectic Poisson structure on this partial compactification, whose fibers are isomorphic to regular Hessenberg varieties.