Masaki Kashiwara

This result was announced in [K]. Mebkout [Me] gave another proof to this theorem. 0. 2. In [KK], we have already shown that J x gives an equivalence and that DR X is fully faithful; i.e. for any Jt\ -^"' e Dj h ( ^x) > we have 0.3. (Sflf-^dim X], and *:D b (X)->D b (X) by x Here o denotes the opposite category and Q x denotes the sheaf of differential forms with the highest degree. Then we have DR X * = *DR X and **=id. 0. 4. Now for ^'eD? h (^z) 5 we put F'=DR X W)*. Then in [KK] we have prove

We study the properties of level-zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by T. Akasaka and the author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level-zero fundamental weight as an extremal weight is irreducible,

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan-Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the pe

The first one is [6] and the second one is [8].(*#) g (resp., J^) denotes the sheaf of micro-differential (resp.linear differential) operators of finite order.See also the list of notations given at the end of this section.Example.Let us consider the following ordinary differential equation:If a 7*0, (0.1) is clearly an equation with irregular singularities.Now consider the following correspondences (0.2) and (0.3).CO 00© (^m/^m-i) is Noetherian.It is easy to verify that © (^m/^m-i) is a m=0 m=0