ABSTRACT
This chapter considers sl2 action on the natural, not the adjoint, representation of g; assumes that g is classical to speak of its natural representation. It follows from and Weyl's Theorem on complete reducibility that any finite-dimensional representation of sl2 is determined up to isomorphism by a partition. The chapter parametrizes nilpotent orbits by partitions, and classifies nilpotent orbits in the exceptional algebras in Chapter 8 by an entirely different method. It begins by stating the main results on the parametrization of nilpotent orbits and then indicates a cleaner statement that can be made at the expense of introducing the full orthogonal groups. The chapter constructs (more or less) canonical representatives of conjugacy classes of standard triples in the classical algebras. These provide us with a normal form representative of each nilpotent orbit and enable us to compute its weighted Dynkin diagram. The chapter shows how to compute the weighted Dynkin diagram for an arbitrary classical algebra g.
