Black Hole Nilpotent Orbits and Tits Satake Universality Classes

In this paper we consider the problem of classification of nilpotent orbits for the pseudo-quaternionic coset manifolds U/H* obtained in the time-like dimensional reduction of N = 2 supergravity models based on homogeneous symmetric special geometries. Within the D=3 approach this classification amounts to a classification of regular and singular extremal black hole solutions of supergravity. We show that the pattern of such orbits is a universal property depending only on the Tits-Satake universality class of the considered model, the number of such classes being five. We present a new algorithm for the classification and construction of the nilpotent orbits for each universality class which is based on an essential use of the Weyl group W of the Tits Satake subalgebra U_{TS} of U and on a certain subgroup W_H thereof. The splitting of orbits of the full group Uinto suborbits with respect to the stability subgroup H* is shown to be governed by the structure of the discrete coset W/W_H. For the case of the universality class SO(4,5) /[SO(2,3) x SO(2,2)] we derive the complete list of nilpotent orbits which happens to contain 37 elements. We also show how the universal orbits are regularly embedded in all the members of the class that are infinite in number. As a matter of check we apply our new algorithm also to the Tits Satake class G_(2,2)/[SL(2)x SL(2)] confirming the previously obtained result encompassing 7 nilpotent orbits. Perspectives for future developments based on the obtained results are outlined.