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 $G\mathrm{/}{P}_S$ and $L\mathrm{/}Q$; in particular we classify the $L$-orbits here, and find natural explicit integral transforms between degenerate principal series of $L$ and $G$.