Corrado Marastoni

Radon transforms for quasi-equivariant D-modules on generalized flag manifolds

Abstract In this paper we deal with Radon transforms for generalized flag manifolds in the framework of quasi-equivariant D -modules. We shall follow the method employed by Baston–Eastwood and analyze the Radon transform using the Bernstein–Gelfand–Gelfand resolution and the Borel–Weil–Bott theorem. We shall determine the transform completely on the level of the Grothendieck groups. Moreover, we point out a vanishing criterion and give a sufficient condition in order that a D -module associated to an equivariant locally free O -module is transformed into an object of the same type. The case of maximal parabolic subgroups is studied in detail.

Grassmann duality for D-modules

Abstract We generalize the main results on projective duality (see [2], [4], [12]) to the case of the correspondence between “dual” Grassmann manifolds G and G ∗. The new aspect is that the “incidence variety” S ⊂ G × G ∗ is no longer smooth, a fact which requires the tools of the theory of b-functions ([7], [17]). In particular, we obtain an equivalence between the categories of sheaves on G and G ∗, as well as between those of D -modules; then, quantizing this equivalence, we explicitly calculate the transform of a D -module associated to a holomorphic line bundle.

Quantification de la dualité de Grassmann

Let V be a complex vector space of dimension n, %plane1D;53E; (resp. %plane1D;53E;*) the Grassmann manifold of p-dimensional (resp. (n — p)-dimensional) subspaces of V, and of Ω the relation of transversality in %plane1D;53E;*%plane1D;53E;*. We announced in [6] equivalences between derived categories of sheaves and of D-modules on %plane1D;53E; and %plane1D;53E; defined by the integral transforms associated to Ω. We show here that these transforms exchange the D-modules associated to the holomorphic line bundles on %plane1D;53E; and %plane1D;53E;*. This is equivalent to “quantizing” the underlying contact transformation between certain open dense subsets of the cotangent bundles. In the case p = 1, we recover already known results for the projective duality (see [1] and [5]).

Generalized Verma modules, b-functions of semi-invariants and duality for twisted D-modules on generalized flag manifolds

Let G be a connected semisimple algebraic group over C, P a parabolic subgroup, g and p their Lie algebras. We prove a microlocal version of Gyojas conjectures [2] about a relation between the irreducibility of generalized Verma modules on g induced from p and the zeroes of b-functions of P-semi-invariants on G. Our method uses a duality for twisted D-modules on generalized flag manifolds. To cite this article: C. Marastoni, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 111–116.