Seminar Algebra and Geometry: Lucas Fresse (Hebrew University of Jerusalem)

On some geometric properties of orbital varieties

To a nilpotent element x in a reductive Lie algebra, one can attach several algebraic varieties which play roles in representation theory: its nilpotent orbit; the intersection of its nilpotent orbit with a Borel subalgebra (the irreducible components of this intersection are called orbital varieties); the fiber over x of the Springer resolution. There is a close relation between the Springer fiber over x and the orbital varieties attached to x. In this talk, we rely on this relation in order to study two properties of orbital varieties: the smoothness, and the property to contain a dense B-orbit. We concentrate on type A. We provide several criteria which suggest that the two mentioned properties are related. This is a joint work with Anna Melnikov.

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