Nilpotent orbits in real symmetric pairs and stationary black holes

In the study of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determine the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of (SL₂(R))⁴ acting on the fourth tensor power of the natural 2-dimensional SL₂(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.