Bernstein-Gelfand-Gelfand Sequences of Invariant Operators
by
Lukáš Krump
Charles University, Prague

Poster

Parabolic structures on manifolds have been studied intensively in recent years. The concept of parabolic geometries includes many classical geometrical structures on general manifolds, starting with the conformal and quaternionic structures, CR-structures, contact structures, etc. A parabolic structure is given by a fibre bundle whose structure group P is a parabolic subgroup in a simple group G. The Lie algebra g of G has then to be k-graded. There is supposed also the existence of a Cartan connection on the manifold.

One of the important problems in this area is understanding of invariant operators on manifolds with such geometries. There are constructions that allow us to describe a broad class of so-called standard operators (of any order). For a given Lie algebra, the operators are constructed as projections of associated bundles to certain representation spaces onto irreducible components. It turns out that many properties of the operators then can be read out purely from finite-dimensional representation-theoretical data. The method can then be extended to the case of non-standard operators.

A natural question is a systematization of standard (and non-standard) operators. We consider sequences of operators - so-called BGG (Bernstein-Gelfand-Gelfand) sequences. The BGG sequences are usually described by means of Hasse diagram. It determines the form (shape) of the BGG sequence. There are classical ways of constructing these sequences. In the talk will be mentioned a new method of construction of the Hasse diagram and of the BGG sequences directly from the so-called weight graph of the positive part of the grading of the given Lie algebra. This gives in fact the same set of operators, and also the individual data (weights of representations and orders of operators) in the sequence (for given initial data) are computed more directly.

A practical recipe for construction of BGG sequences in particular cases will be shown together with the resulting sequences for interesting instances of parabolic structures. There exists also a formula for the orders of resulting operators.

Date received: June 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-27.