Journal of Lie Theory 18 (2008), No. 4, 757--774
Copyright Heldermann Verlag 2008
Generalized Dolbeault Sequences in Parabolic Geometry
Peter Franek
Institut of Mathematics, Charles University, Sokolovska 83, 18675 Prague, Czech Republic
franp9am@artax.karlin.mff.cuni.cz
[Abstract-pdf]
\def\R{{\Bbb R}} We show the existence of a sequence of invariant differential operators on a particular homogeneous model $G/P$ of a Cartan geometry. The first operator in this sequence is closely related to the Dirac operator in $k$ Clifford variables, $D=(D_1,\ldots, D_k)$, where $D_i = \sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\SS)\to C^\infty((\R^n)^k, \SS)$. We describe the structure of these sequences in case the dimension $n$ is odd. It follows from the construction that all these operators are invariant with respect to the action of the group $G$. These results are obtained by constructing homomorphisms of generalized Verma modules, which are purely algebraic objects.
Keywords: Dirac operator, parabolic geometry, BGG, generalized Verma module.
MSC: 58J10, 34L40
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