Chapter 2 - Orbital Varieties, Goldie Rank Polynomials and Unitary Highest Weight Modules

Abstract

This chapter briefly explains orbital varieties, Goldie rank polynomials, and unitary highest weight modules. The Goldie rank polynomials are intimately related to the geometry of the nilpotent orbits via what was at first just a strange coincidence with the Springer theory. However, one can now directly attach to a nilpotent orbit polynomials that are not quite the same, but that span the same space. These are the characteristic polynomials of the orbital varieties attached to a given orbit. It turns out that these are given by exactly the same formulae, but involving geometric analogs of the Kazhdan-Lusztig polynomials that can be and are different. This gives an entirely new twist to the orbit method and insight into geometry from representation theory rather than vice versa. The relationship between Goldie rank and characteristic polynomials allows one to conclude that the latter (and hence the former) can be expressed as a positive linear combination of products of (distinct) positive roots.