| To any holonomic D-module on an abelian variety one may attach an algebraic group in a natural way, using a Tannakian description for the convolution product. This allows to study subvarieties of abelian varieties via the groups for the corresponding intersection cohomology D-module. For example, on the moduli space of principally polarized abelian varieties, the groups for the theta divisor cut out interesting strata that refine the Andreotti-Mayer stratification and conjecturally characterize the loci of Jacobian varieties, intermediate Jacobians etc. We are then led to various general questions: What information on a subvariety is encoded in the corresponding group? How can one determine these groups? And can their representation theory be used as a tool in geometry? The first part of this talk will give a motivated introduction to the Tannakian formalism for abelian varieties, starting from a generalization of the Green-Lazarsfeld generic vanishing theorem. We will then discuss recent results that relate the arising Weyl groups to the monodromy of Gauss maps and point towards a first possible answer to the above questions. |