FWF-Project P27072-N25

New directions in the theory of BGG sequences

Individual research project funded by the Austrian Science Fund ("Fonds zur Förderung der wissenschaftlichen Forschung" - FWF)

Project leader: Andreas Cap, Faculty of Mathematics, University of Vienna.

People supported by the project:

Main international collaborators:

Scientific Aims: The BGG-sequences (short for Bernstein-Gelfand-Gelfand-sequences studied in this project are sequences of differential operators intrinsic to certain geometric structures. Under certain types of flatness assumptions they are known to be complexes respectively to contain certain subcomplexes. The name draws from a duality relating these sequences to homomorphisms of generalized Verma modules, which form Lepowski's generalizations of the Bernstein-Gelfand-Gelfand resolutions of finite dimensional representations of semi-simple Lie algebras.

After some initial constructions, in particular for the case of conformal structures, a general theory of BGG sequences in the setting of so-called parabolic geometries was developed around the year 2000. Since then, a large number of geometric applications of BGG sequences have been found and they are among the central tools for the theory of parabolic geometries. In particular, there is a close connection between BGG sequences and overdetermined systems of PDEs which are intrinsic to parabolic geometries. Among the solutions to these systems, there is a special subclass of normal solutions, which give rise to holonomy reductions of parabolic geometries, for which a general theory has been developed recently.

The basic aim of the project is to apply the techniques that have led to the construction of BGG-sequences for the developement of new tools which can be applied to geometric problems beyond the realm of parabolic geometries. The four main directions of study planned for the project are:

  • Geometric compactifications and holonomy reductions: This is a joint project with Rod Gover, which continues the direction of holonomy reductions of Cartan geometries described above. A crucial feature of such holonomy reductions (which is not present in the case of reductions of principal connections) is that they lead to a decomposition of the underlying manifold into strata of different dimensions, which inherit different geometric structures. This gives rise to notions of compatibility of geometric structures on manifolds of differnt dimensions, which in particular can be applied to the case of structures on the interior and on the boundary of a manifold with boundary. Weakening the conditions needed for a holonomy reduction one arrives at notions of geometric compactifications which should be interesting for several fields in mathematics (e.g. scattering theory) and theoretical physics (e.g. general relativity and the AdS/CFT-correspondence).

  • Relative BGG-sequences: This is a joint project with Vladimir Soucek aiming at the construction of a relative version of BGG- sequences, which are assoicated to a pair of nested parabolic subalgebras rather than a single parabolic subalgebra. On the one hand, this should lead to a conceptual construction of invariant differential operators beween natural bundles induced by representations of singular infinitesimal charcter, which are not accessible for standard BGG-sequences. On the other hand, such sequences can be used to resolve certain sheafs, offering a starting point for curved versions of Penrose transforms.

  • Pushing down BGG sequences to leaf spaces: This joint project with T. Salac is motivated by ideas from Clifford Analysis and a recent result on integral geometry on complex projective space by M. Eastwood and H. Godschmidt. The broad idea is to consider foliations defined by certain infinitesimal automorphisms parabolic geometries. A local leaf-space for such a foliation should inherit a geometric strucuture and BGG sequcences should give rise to sequences of differential operators on such a leaf-space which are intrinsic to this structure.

  • Poisson transforms: This part of the project is centered around the thesis project of Christoph Harrach. The main idea is that, generalizing work of P.Y. Gaillard, one can define a Poisson transform mapping differential forms on the boundary of a rank one symmetric space to differential forms on the symmetric space itself. To do this, one needs certain invariant differential forms, which can be described in terms of finite dimensional representation theory. This description leads to an efficient way to design the properties of a transform. There is hope that some of these constructions admit generalizations to curved settings, for example in the setting of Poincaré-Einstein manifolds.

    Publications related to the project:

    Talks related to the project: