Nilpotent orbits and theta-stable parabolic subalgebras

Abstract:

In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra ${\mathfrak {g}}$ under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on ${\mathfrak {p}}_{{}_{\mathbb {C}}}$, the “complex symmetric space associated with ${\mathfrak {g}}$”. The Kostant-Sekiguchi correspondence, a bijection between nilpotent orbits in ${\mathfrak {g}}$ and nilpotent orbits in ${\mathfrak {p}}_{{}_{\mathbb {C}}}$, is also used. We identify a fundamental set of noticed nilpotents in ${\mathfrak {p}}_{{}_{\mathbb {C}}}$ and show that they allow us to recover all other nilpotents. Finally, we study the behaviour of a principal orbit, that is an orbit of maximal dimension, under our classification. This is not done in the other classification schemes currently available in the literature.