Action of Hecke algebra on the double flag variety of type AIII

Consider a connected reductive algebraic group $G$ and a symmetric subgroup $K$. Let $\mathfrak{X}=K\mathrm{/}{B}_K\times G\mathrm{/}P$ be a double flag variety of finite type, where $B_K$ is a Borel subgroup of $K$, and $P$ a parabolic subgroup of $G$. A general argument shows that the orbit space $\mathbb{C}\mathfrak{X}\mathrm{/}K$ inherits a natural action of the Hecke algebra $\mathcal{H}=\mathcal{H}(K,B_K)$ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem. In this paper, we determine the explicit action of $\mathcal{H}$ on $\mathbb{C}\mathfrak{X}\mathrm{/}K$ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $\mathbb{C}\mathfrak{X}\mathrm{/}K$ as a direct sum of induced representations.