Combinatorics Related to Orbit Closures of Symmetric Subgroups in Flag Varieties

Let G be a connected reductive group over an algebraically closed field k of characteristic not 2; let θ ∈ Aut(G) be an involution and K = Gθ ⊆ G the fixed point group of θ and let P ⊆ G be a parabolic subgroup. The set K\G/P of (K , P)-double cosets in G plays an important role in the study of Harish Chandra modules. In [BH00] we gave a description of the orbits of symmetric subgroups in a flag variety G/P mainly using geometric arguments. For general P , it is difficult to describe the combinatorics of the decomposition of the closure of a double coset in terms of K × P double cosets. However, in some special cases one can describe the combinatorics of the closures of the double cosets inmore detail. This paper discusses the special case that P contains a θ -stable Levi factor L and the set of roots of the connected center S of L is a root system with Weyl group W (S) = NG(S)/ZG(S). Here NG(S) (resp. ZG(S)) is the normalizer (resp. centralizer) of S in G. In this case the combinatorics of the Weyl Group can be used to describe the closures of the double cosets of a part of the double coset space which includes the open and closed orbits and we get a number of results similar to the case that P = B a Borel subgroup. This root system condition on P is satisfied in many cases. For example in the case that P is a minimal parabolic k0-subgroup of G or a minimal θ -split parabolic subgroup of G or a minimal (θ, k0)-split parabolic k0-subgroup of G. Here k0 ⊆ k is a subfield of k and G, θ are defined over k0.