Relative BGG sequences; II. BGG machinery and invariant operators

Abstract

For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on such bundles. We define a (rather weak) notion of "compressability" for operators acting on relative differential forms with values in a relative tractor bundle. The we develop a general machinery which converts a compressable operator to an operator on bundles associated to completely reducible representations on relative Lie algebra homology groups. Applying this machinery to a specific compressable invariant differential operator of order one, we obtain a relative version of BGG (Bernstein--Gelfand--Gelfand) sequences. All our constructions apply in the case $P=G$, producing new and simpler proofs in the case of standard BGG sequences. We characterize cases in which the relative BGG sequences are complexes or even fine resolutions of certain sheaves and describe these sheaves. We show that this gives constructions of new invariant differential operators as well as of new subcomplexes in certain curved BGG sequences. The results are made explicit in the case of generalized path geometries.