FWF-Project P33559-N

Applications of parabolic geometries and 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 in Vienna related to the project:

Main international collaborators:

Scientific Aims: Parabolic geometries are a class of geometric structures, which in their standard descriptions look very diverse. They do admit a uniform description as Cartan geometries with homogeneous model the quotient of a (real or complex) semisimple Lie group by a parabolic subgroup. This description already indicates a strong connection to semi-simple representation theory, which indeed plays a major role in the study of parabolic geometries. Combined with geometric methods tailored to the specific situation of second order geometric structures, this leads to a large number of very efficient tools for the study of parabolic geometries, and the general theory of these structures is developed very well by now.

One of the core tools for parabolic geometries is provided by Bernstein-Gelfand-Gelfand sequences (BGG sequences). These are sequences of higher order invariant differential operators intrinsically associated to any parabolic geometry. They can be viewed as curved generalizations of the resolution of finite dimensional irreducible representations by generalized Verma modules which was obtained by J. Lepowsky as a generalization of the original BGG resolution by Verma modules. While the theory of BGG sequences dates back to the early 2000's, a substantial extension of the theory was recently obtained in joint work of A. Cap and V. Soucek in the form of relative BGG sequences.

The basic aim of the project is to develop the theory of parabolic geometries and BGG sequences and to apply them to geometric problems beyond the realm of parabolic geometries. The five main directions of study planned for the project are:

  • Geometric compactifications and holonomy reductions: This is a continuation of a long term joint project of A. Cap and A.R. Gover, based on the theory of holonomy reductions of Cartan geometries that we developed in joint work with M. Hammerl. These holonomy reductions turn out to provide model cases for various types of geometric compactifications generalizing R. Penrose's concept of conformal compactness. This provides interesting connections to several areas of mathematics (e.g. scattering theory) and theoretical physics (e.g. general relativity and the AdS/CFT-correspondence). In particular, we plan to apply tractor methods to the study of the mass of asymptotically hyperbolic metrics and generalizations of this. This research also admits applications to the study of compactifications of symmetric spaces, which gives connections to the research on Poisson transforms described below.

  • Poisson transforms: This part of the project builds on joint work with P. Julg and on C. Harrach's thesis. Generalizing work of P.Y. Gaillard, we construct Poisson transforms for differential forms that are adapted to flat parabolic geometries on the boundary (or on boundary components) of certain symmetric spaces. To a large extent, these questions can be reduced to the construction of invariant differential forms which can be done using finite dimensional representation theory. The question of whether the transform produces harmonic forms turns out to be closely related to the question whether it descends from a (twisted) de-Rham sequence to the corresponding BGG sequence on the boundary. The main motivation for these questions is applications to KK-theory for which parts of specific BGG sequences and Poisson transforms have to be combined to form special complexes that generate KK-groups. This research has obvious connections to the study of geometric compactifications. Moreover, there is hope that some of these constructions admit generalizations to curved settings, for example in the setting of Poincaré-Einstein manifolds.

  • BGG-machinery and applications: This part of the project builds on the recent joint work with V. Soucek in which we showed that there is a relative version of the BGG machinery associated to a pair of nested parabolic subalgebras. While the machinery used to construct relative BGG sequences is similar to the original BGG construction, the resulting sequences are of different nature. In particular, one obtains complexes that are resolutions of certain sheaves without assumptions (or with only weak assumptions) on local flatness. While the general theory of these sequences has been established in our joint articles, this theory has not been worked out in detail in any case. Studying this for two structures of broader interest is the content of the thesis projects of Z. Guo and M. Wasilewicz. Related to this is the question of developing a relative version of tractor calculus, which in turn can be used to study the full tractor calculus for these structures, which is not completely understood for the structures in question. A second aim for this part of the project is to generalize the construction of invariant operators in singular infinitesimal character that is provided by the relative BGG machinery to the case of maximal parabolics. This will be studied jointly with V. Soucek.

  • The bundle of Weyl structures and non-linear invariant PDE: This a joint project with T. Mettler. Motivated by earlier work by M. Dunajski and T. Mettler we have obtained a general construction that associated to any $|1|$-graded parabolic geometry a certain bi-Lagrangean structure on the total space of a natural fiber bundle. It turns out that sections of that bundle can be identified with Weyl structures for the corresponding parabolic geometry. This opens the possibility to study Weyl structures via submanifold geometry in a bi-Lagrangean geometry. The motivation for this is that special Weyl structure turn out to be related to non-linear invariant PDE of Mongé-Ampère type. In low dimensions, the latter turn out to be closely related to convex projective structures, which provides a very promising relation to topics like character varieties and higher Teichmüller theory. This opens the possibility for completely new applications of parabolic geometries.

  • Geometry of differential equations: It is well known that path geometries, which can be equivalently described as a parabolic geometry, provide an equivalent description of systems of second order ODE. It has been known for quite some time that there is an analogous description of systems of higher order ODE in terms of a canonical Cartan geometry (not of parabolic type). In recent joint work with B. Doubrov and D. The, we have obtained a new construction of this canonical Cartan geometry, which brings it into very similar form to parabolic geometries. In particular, this opens the possibility to generalize parts of the (relative) BGG machinery to the setting of higher order ODE. Studying this and applications to ODE theory, e.g. on existence of geometric structure on spaces of solutions, will be an important part of the project.

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