Some properties of orbital varieties in extremal nilpotent orbits

Lucas Fresse (1)

1 IECL - Institut Élie Cartan de Lorraine

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 sub-regular 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.

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]; Théorie des représentations [math.RT]

Lucas Fresse. Some properties of orbital varieties in extremal nilpotent orbits. Representations and nilpotent orbits of Lie algebraic systems, 330, Birkhäuser/Springer, pp.305-329, 2019, Progress in Mathematics. ⟨hal-01739780⟩