Nilpotent orbit of real symmetric pair

Let g = g0 ⊕ g1 be a Z/2Z-graded real semisimple Lie algebra. Then (g, g0) is called a real symmetric pair. An x ∈ g1 is called nilpotent if the adjoint map adx : g → g is nilpotent. We let G0 be a Lie group with Lie algebra g0 acting on g1. The problem is to determine the orbits of G0 on the set of nilpotent elements of g1. We wil show several algorithmic techniques that help with solving this problem. The methods will be illustrated in an example where g is the split real form of the Lie algebra of type D4, and the action of G0 on g1 is isomorphic to the action of SL(2, R) 4 on the tensor product of four copies of R 2 . The nilpotent orbits in this example are of interest in theoretical physics, in particular in the study of black holes. (This is joint work with Heiko Dietrich, Daniele Ruggeri, and Mario Trigiante.)