Birational sheets in linear algebraic groups
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 nitely-many subvarieties of g consisting of equidimensional orbits, called birational sheets: their de nition is less immediate than the one of a sheet, but they enjoy better geometric and representation-theoretic properties and are central in Losev's proposal to give an Orbit method for semisimple Lie algebras.
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 simply-connected.
We conclude with an overview of the main features of these varieties, which mirror some of the properties of the objects defined by Losev.
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