ABSTRACT
This chapter describes about the nilpotent orbits of the adjoint group (G) on a semisimple Lie algebra(g). The nilpotent orbits are not parametrized by sets of partitions in exceptional types. The chapter considers a way to list the nilpotent orbits using the carrier algebras. Corollaryl describe a simple algorithm for deciding whether a given weighted Dynkin diagram corresponds to a nilpotent orbit. Using that algorithm only straightforward to obtain the classification of the nilpotent orbits in the exceptional cases. The chapter introduces the theory of ?-groups, which were introduced by Vinberg in the 1970's. They form a class of representations of reductive algebraic groups, sharing many properties of the adjoint representation of a semisimple algebraic group. In particular, the strata of the nullcone are orbits that are consequently finite in number.
