Smooth Orbital Varieties and Orbital Varieties with a Dense B-Orbit

Abstract

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of a reductive Lie algebra. Orbital varieties arise in representation theory as the geometric analogs of the enveloping algebra’s primitive ideals. They are also related to the irreducible components of the Springer fibers. In type A, the orbital varieties have a convenient parameterization by standard Young tableaux, in particular, one has a natural duality on the set of orbital varieties induced by the Young tableau transposition. In this paper, we provide several combinatorial criteria to guarantee that an orbital variety is smooth or contains a dense orbit for the action of the Borel subgroup B. We point out that, in all the cases that we can compute, the smooth orbital varieties and the orbital varieties which have a dense B-orbit correspond via the aforementioned duality.