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BGG sequences on spheres. (English) Zbl 1037.43016
The main topic of the paper is a study of properties of (standard) conformally invariant differential operators on the sphere \(S^n\) with its standard conformal structure. Such operators are appearing in certain complexes of differential operators similar to the classical de Rham complex of differential forms. They are usually called Bernstein-Gelfand-Gelfand (BGG) complexes. To every irreducible representation of the conformal group of the sphere, there is one such BGG sequence. Invariant differential operators are acting between sections of fibre bundles associated to (irreducible) representations of the group \(C(n).\) The spaces of sections of such bundles have the structure of \(\text{Spin}(n+1)\)-modules and their decomposition into irreducible \(Spin(n)\)-modules is given by the corresponding branching rules. These simple representation theoretical tools are used to study properties of BGG sequences on the sphere and properties of spaces of solutions of individual invariant operators in the sequence (in particular, properties of solutions of the corresponding twistor operators). In the paper, it is shown that BGG sequences on the sphere always form a complex. This is a simple consequence of the branching rules for the source and the targets of individual operators in the sequence and it needs no specific information on the properties of the operators themselves. Useful necessary and sufficient conditions for the exactness of the complex are described. Full information on spaces of solutions (i.e. their decomposition into \(Spin(n)\)-irreducible components), on spaces of solutions of individual operators in the BGG sequence are obtained. In particular, it applies to the case of twistor operators. The spaces of local solutions of twistor operators are hence determined for every BGG sequence, because they coincide with the space of global solutions on the spheres.
MSC:
43A85 Harmonic analysis on homogeneous spaces
22E46 Semisimple Lie groups and their representations
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