Components of Springer fibers associated to closed orbits for the symmetric pairs (Sp(2n),GL(n)) and (O(n),O(p)×O(q)) I

Let (G,K) be either (Sp(2n),GL(n)) or (O(n),O(p)×O(q)), p+q=n. An explicit geometric description of components of Springer fibers associated to closed K-orbits in the flag variety of G is given. This description is used to give a simple algorithm for computing associated cycles of discrete series representations for the groups Sp(2n,R) and SOe(p,q).