ABSTRACT
This chapter shows how to write down all the nilpotent orbits in any semisimple Lie algebra g in terms of data easily computed from its Dynkin diagram. It follows the approach of Bala and Carter. The chapter looks at nice subalgebras of g meeting a given nilpotent orbit. It is natural to divide nilpotent orbits of g into two classes: those that meet regular reductive subalgebras and those that do not. The former can be classified by doing calculations in smaller algebras, and it turns out that there are sufficiently few of the latter to be tractable. Bala and Carter's fundamental insight was that such a parametrization can be obtained if one repeats the above program using Levi subalgebras (which are regular and reductive) instead of arbitrary regular reductive subalgebras. The chapter gives four pieces of information about every nilpotent orbit: its Bala-Carter label, weighted diagram, dimension, and fundamental group.
