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Birational sheets of conjugacy classes in reductive groups

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If G is an algebraic group acting on a variety X, the sheets of X are the irreducible components of subsets of elements of X with equidimensional G-orbits. For G complex connected reductive, the sheets for the adjoint action of G on its Lie algebra g were studied by Borho and Kraft in 1979. More recently, Losev has introduced finitely-many subsets of g consisting of equidimensional orbits, called birational sheets: their definition is not as immediate as the one of a sheet, but birational sheets behave better in geometric and representation-theoretic terms. Indeed, birational sheets are disjoint, unibranch varieties with smooth normalization, while this is not true for sheets, in general. Moreover, the G-module structure of the ring of functions C[O] does not change as the orbit O varies in a birational sheet. In this seminar, we define an analogue of birational sheets of conjugacy classes in G: we start by recalling Lusztig-Spaltenstein induction of conjugacy classes in terms of the so-called Springer generalized map and analyse its interplay with birationality. With this tools, we give a definition of birational sheets of G in the case that the derived subgroup of G is smplyconnected. We conclude with an overview of the main features of these varieties, which mirror some of the properties enjoyed by the objects defined by Losev.

This talk is part of the Algebra seminar series.

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