Bernstein-Gelfand-Gelfand sequences

TL;DR

This paper constructs invariant differential operator sequences for geometries modeled on homogeneous spaces G/P, generalizing Bernstein-Gelfand-Gelfand resolutions to both smooth and holomorphic categories, with applications to curved geometries.

Contribution

It introduces a new method to build invariant differential operator sequences for parabolic geometries, extending BGG resolutions to curved cases using elementary representation theory.

Findings

Constructed sequences of invariant differential operators for G/P geometries.

Extended BGG resolutions to curved geometries with explicit cohomology relations.

Presented a comprehensive theory of these geometric structures for the first time.

Abstract

This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and $P\subset G$ is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invariant differential operators for such geometries, both in the smooth and the holomorphic category. For G simple, these sequences specialize on the homogeneous model G/P to the celebrated (generalized) Bernstein-Gelfand-Gelfand resolutions in the holomorphic category, while in the smooth category we get smooth analogs of these resolutions. In the case of geometries locally isomorphic to the homogeneous model, we still get resolutions, whose cohomology is explicitly related to a twisted de Rham cohomology. In the general (curved) case we get distinguished curved analogs of all the invariant differential operators occurring in Bernstein-Gelfand-Gelfand resolutions (and their smooth analogs). On the way to these results, a significant part of the general theory of geometrical structures of the type described above is presented here for the first time.