Seminari del Dipartimento di Matematica

Seminario del 2020

Filippo Ambrosio

Birational sheets in linear algebraic groups

nel ciclo di seminari: ALGEBRA E GEOMETRIA

algebra e geometria

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 subvarieties of g consisting of equidimensional orbits, called birational sheets: their definition 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 the first part of the seminar we give an historical overview on sheets and recall some basics about algebraic groups and Lusztig-Spaltenstein induction in terms of the so-called Springer generalized map and analyse its interplay with birationality. This will allow us to introduce Losev's birational sheets. The last part is aimed at defining an analogue of birational sheets of conjugacy classes in G, under the hypothesis that the derived subgroup of G is simply connected. We will conclude with an overview of the main features of these varieties, which mirror some of the properties enjoyed by the objects defined by Losev.