ABSTRACT
For the purpose of contrast and comparison with the nilpotent case, this chapter looks at the classification of semisimple orbits in g. It emphases on understanding the centralizer structure for semisimple elements, and then classifies the semisimple orbits in a reductive Lie algebra. The classification begins with a construction which ultimately produces all Cartan subalgebras. Semisimple orbits are parametrized by points in a fundamental domain for the action of the Weyl group on a Cartan subalgebra. In particular, there are infinitely many semisimple orbits. The chapter concludes with a brief discussion of the topology of semisimple orbits. It should be emphasized that many of the ideas are needed in the study of nilpotent orbits. Aside from the fact that semisimple orbits are rather easily parametrized, they also may be topologically characterized among all adjoint orbits.
