Quantization of Grassmann duality

Let V be a complex vector space of dimension n, G (resp. G*) the Grassmann manifold of p-dimensional (resp. (n-p)-dimensional) subspaces of V, and of Omega the relation of transversality in G x G*. In a previous note we announced in equivalences between derived categories of sheaves and of D-modules on G and G* defined by the integral transforms associated to Omega. We show here that these transforms exchange the D-modules associated to the holomorphic lines bundles on G and G*. This is equivalent to ''quantizing'' the underlying contact transformation between certain open dense subsets of the contangent bundles. In the case p = 1, we recover already known results for the projective duality.