Conjugacy classes of parabolic subalgebras in complex semi-simple lie algebras

For a complex semi simple Lie algebra g, Richardson's dense orbit theorem gives a map between conjugacy classes of parabolic subalgebras in g and conjugacy classes of nilpotent elements. Unfortunately, this map is not surjective, in general, and hence does not give a direct classification of the nilpotent conjugacy classes in g. Despite this, the theorem is used by Bala and Carter to produce an indirect classification of the nilpotent conjugacy classes in g. The map is not injective, either, and this thesis attempts to discover a necessary and sufficient condition for two parabolic subalgebras to give the same nilpotent conjugacy class under the above map. Springer conjectured that associated parabolics would give the same nilpotent conjugacy class. The problem was also raised in another form by Dixmier in his work concerning the distribution of nilpotent polarisable elements in g. He conjectured a generalisation of Kostant's results on the regular nilpotent elements. We prove both these conjectures correct, the method of proof being inspired by Dixmier's work. Unfortunately, the necessary and sufficient condition is clearly more complicated than this and we give two examples (one trivial, one non-trivial) of non-associated parabolics giving the same nilpotent conjugacy class under Richard son's map