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

For the pairs of complex reductive groups (G,K)=(Sp(2n),Sp(2p)×Sp(2q)) and (SO(2n),GL(n)) components of Springer fibers associated to closed K-orbits in the flag variety B of G are described. The closed K-orbits in B correspond to discrete series representations of and SO∗(2n). We give an algorithm to compute the associated variety, the closure of a nilpotent K-orbit K⋅f, of each discrete series representation and we describe the structure of the corresponding component of the Springer fiber μ−1(f). The description of these components has applications to the computation of associated cycles of discrete series representations; this is the topic of the sequel to the present article.