Some Properties of Orbital Varieties in Extremal Nilpotent Orbits

Abstract

The intersection between a nilpotent orbit of a simple Lie algebra and a Borel subalgebra is always equidimensional. Its irreducible components are called orbital varieties. Orbital varieties belonging to different nilpotent orbits may have quite different behaviours. The orbital varieties of the subregular nilpotent orbit are always smooth but they have in general infinitely many B-orbits. At the opposite, the minimal nilpotent orbit is spherical but its orbital varieties may have singularities. In this paper, we characterize the orbital varieties of the subregular nilpotent orbit which have a finite number of B-orbits and we give a smoothness criterion for the orbital varieties of the minimal nilpotent orbit.