Real double coset spaces and their invariants

Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H=Gσ denote the fixed group of σ and let K=Gθ denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.