On the conormal bundle of a G-stable subvariety

Abstract:

Let G be a reductive algebraic group and X a smooth G-variety. For a smooth locally closed G-stable subvariety M⊂X, we prove that the G-complexity of the (co)normal bundle of M is equal to the G-complexity of X. In particular, if X is spherical, then all (co)normal bundles are again spherical G-varieties. If X is a G-module with finitely many orbits, the closures of the conormal bundles of the orbits coincide with the irreducible components of the commuting variety. We describe properties of these closures for the representations associated with short gradings of simple Lie algebras.