Lagrangian Grassmannians: Vector Bundles and Representation Theory

Modern algebraic geometry identifies the derived category as one of the main invariants of an algebraic variety. In some cases, the derived category can be described in terms of linear algebra: when one can find in it a so-called full exceptional collection (this approach goes back to the works of A. Beilinson and I. M. Gelfand). An old conjecture states that such a collection can be found for all rational homogeneous varieties. In the talk we will explain what is known about the derived categories of Lagrangian Grassmannians, in particular, in terms of representations of the corresponding non-reductive parabolic subgroup. Through this example, we will try to explain all the emerging terms, objects and constructions.