On generalized Steinberg theory for type AIII

The multiple flag variety X = Gr(Cp+q, r) x (Fl(Cp) x Fl(Cq)) can be considered as a double flag variety associated to the symmetric pair (G, K) = (GLp+q(C), GLp(C) x GLq(C)) of type AIII. We consider the diagonal action of K on X. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations. In [5], we defined two generalized Steinberg maps from the K-orbits of X to the nilpotent K-orbits in k and those in the Cartan complement of k, respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson-Schensted correspondence.