Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.