Deformation quantization modules on complex symplectic manifolds

We study modules over the algebroid stack $\text{W}[\text{stx}]$ of deformation quantization on a complex symplectic manifold $\text{stx}$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\text{W}[\text{stx}]$-modules, finiteness and duality for the composition of ``good’’ kernels. As a corollary, we get that the derived category of good $\text{W}[\text{stx}]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this framework.