Regular Orbits of Symmetric Subgroups on Partial Flag Varieties

The main result of the current paper is a new parametrization of the orbits of a symmetric subgroup K on a partial flag variety P . The parametrization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of our motivations, as explained below, is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits. In more detail, suppose G is a complex connected reductive algebraic group and let θ denote an involutive automorphism of G. Write K for the fixed points of θ, and P for a variety of parabolic subalgebras of a fixed type in g, the Lie algebra of G. Then K acts with finitely many orbits on P , and these orbits may be parametrized in a number of ways (e.g. [M], [RS], [BH]), each of which may be viewed as a generalization of the classical Bruhat decomposition. (This latter decomposition arises if G = G1 × G1, θ interchanges the two factors, and P is taken to be the full flag variety of (pairs of) Borel subalgebras.) We give our parametrization of K\P in Corollary 2.14 and then turn to applications and examples in later sections. As mentioned above, one of the applications we have in mind concerns the connection with nilpotent coadjoint orbits for K. To each orbit Q = K · p of parabolic subalgebras in P , we obtain such a coadjoint orbit as follows. Let k denote the Lie algebra of K, and consider