Abstract

The talk is based on a work in progress with Vladimir Soucek, focusingon the structure of natural linear differential operators on certain almost Grassmannianmanifolds. As well known, the Verma modules are in fact topological duals ofthe modules of jets of sections of the relevant homogeneous vector bundles. Thus,the structure of homomorphisms of generalized Verma modules reveals the linearinvariant differential operators on the corresponding Klein’s geometries. The Janzen-Zuckermann translation principle proved to be a great tool to understand this, ingeneral. A straightforward algebraic generalization yields the semi-holomic Vermamodules based on the semi-holonomic jets, introduced in the old paper by Mike Eastwoodand myself (more than 25 years back, Journal of Algebra, 1997), and theirlink to curved Cartan geometries was clarified there, aiming at the example of conformalRiemannian geometries. I shall remind those concepts and tools in general,and provide an overview on Grasmannian geometries, focusing mainly on the (3,3)-Grassmannians.