On the existence of smooth orbital varieties in simple Lie algebras

TL;DR

This paper proves that in classical Lie algebras, every nilpotent orbit contains at least one smooth orbital variety, extending known results from type A to other classical types, and discusses exceptions in the exceptional cases.

Contribution

It generalizes the existence of smooth orbital varieties from type A to all classical Lie algebras and provides a method to identify such varieties in exceptional cases.

Findings

Every classical nilpotent orbit has at least one smooth orbital variety.

In exceptional Lie algebras, some nilpotent orbits lack smooth orbital varieties.

A new induction method for orbital varieties is introduced.

Abstract

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (non-exhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.