Symplectic multiple flag varieties of finite type
  Problem: Given a reductive algebraic group G, find all k-tuples of parabolic subgroups (P_1,...,P_k) such that the product of flag varieties G/P_1 x ... x G/P_k has finitely many orbits under the diagonal action of G. In this case we call G/P_1 x ... x G/P_k a multiple flag variety of finite type. (If P_1 is a Borel subgroup, the partial product G/P_2 x ... x G/P_k is a spherical variety.)   In this paper we solve this problem in the case of the symplectic group G = Sp(2n). We also give a complete enumeration of the orbits, and explicit representatives for them. (It is well known that for k=2 the orbits are essentially Schubert varieties.) Our main tool is the algebraic theory of quiver representations. Rather unexpectedly, it turns out that we can use the same techniques in the present case as we did for G = GL(n) in math.AG/9805067. 
Peter Magyar, Jerzy Weyman, Andrei Zelevinsky (Northeastern   University)