Radon–Penrose Transform for D-Modules

Let[formula]be a correspondence of complex analytic manifolds,Fbe a sheaf onX, and M be a coherent DX-module. Consider the associated sheaf theoretical and D-module integral transforms given byΦSF=Rg!f−1F[d] andΦSM=g!f−1M, whereRg!andf−1(resp.gandf−1) denote the direct and inverse image functors for sheaves (resp. for D-modules), andd=dS−dYis the difference of dimension betweenSandY. In this paper, assuming thatfis smooth,gis proper, and (f, g) is a closed embedding, we prove some general adjunction formulas for the functorsΦSandΦS. Moreover, under an additional geometrical hypothesis, we show that the transformationΦSestablishes an equivalence of categories between coherent DX-modules, modulo flat connections, and coherent DY-modules with regular singularities along an involutive manifoldV, modulo flat connections (hereVis determined by the geometry of the correspondence). Applications are given to the case of Penroseʹs twistor correspondence.