Generalized Dolbeault sequences in parabolic geometry. (English) Zbl 1176.17003

Franek, Peter

J. Lie Theory 18, No. 4, 757-773 (2008).

In this article, the author constructs several sequences of homomorphisms between generalized Verma modules of the complex orthogonal Lie algebras. The modules in question have singular infinitesimal character, whence most of the general results on homomorphisms between such modules do not apply to these cases. The motivation for studying these homomorphisms is that via a duality they correspond to invariant differential operators acting between sections of certain homogeneous vector bundles on generalized flag manifolds. The first homomorphism in each sequence is related to a Dirac operator in several variables, and the hope is to obtain resolutions of such Dirac operators which would be very useful in Clifford analysis.
The author proves several general results on homomorphisms between generalized Verma modules and their relations to invariant differential operators. Then he turns to the cases related to Dirac operators, giving an iterative description of the form of the relevant orbits of the affine action of the Weyl group on weights. He proves existence of homomorphisms and determines the orders of the corresponding operators. In some cases, it is shown that the homomorphisms actually form a complex.

Reviewer: Andreas Cap (Wien)

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46	Semisimple Lie groups and their representations
58J10	Differential complexes

Keywords:

generalized Verma module; homomorphism; invariant differential operator; generalized flag manifold