ABSTRACT
This chapter uses general principles to establish the existence of three canonical nilpotent orbits: the principal (or regular) orbit, the subregular orbit, and the minimal orbit. To describe the sense in which these orbits are canonical, the chapter introduces a partial order on the set of nilpotent orbits, which depends on the Zariski closure operation. It shows that the boundary of an orbit (defined to be its closure minus the orbit itself) must have a smaller dimension than the orbit, since it is a Zariski-closed proper subvariety of the closure. The chapter addresses the existence, uniqueness, and properties of nilpotent orbits consisting of subregular elements. It presents a survey of the theory of exponents of a semisimple Lie group. The chapter describes a connection between the cohomology of a complex Lie group, the principal nilpotent orbit, invariant theory, and the formula for the order of a finite Chevalley group.
