On varieties in an orbital variety closure in semisimple Lie algebras

In this note we discuss the closure of an orbital variety as a union of varieties. We show that if semisimple Lie algebra g contains factors not of type An then there are orbital varieties whose closure contains components which are not Lagrangian. We show that the argument does not work if all the factors are of type An and provide the facts supporting the conjecture claiming that if all the factors of g are of type An then the closure of an orbital variety is a union of orbital varieties.