ABSTRACT
In Chapter 5, we showed how to parametrize nilpotent orbits in classical Lie algebras (using partitions), but so far we have not seen how to do this for exceptional algebras. Of course, we know from Chapter 3 that nilpotent orbits correspond bijectively to their weighted Dynkin diagrams, but we still do not know which labelings of the nodes constitute weighted diagrams. In this chapter, we complete the program of Chapter 3 by showing how to write down all the nilpotent orbits in any semisimple Lie algebra g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2571.tif"/> in terms of data easily computed from its Dynkin diagram. We follow the approach of Bala and Carter in [2] and [3]. All the results we state are due to them, though in certain cases we have modified their proofs. The main idea is to look at nice subalgebras of g https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203745809/e79dcd06-dfda-4377-9633-50cbf808d6d2/content/eq2572.tif"/> meeting a given nilpotent orbit.
