Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 18 (2008), No. 4, 757--774Copyright 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 [ Fulltext-pdf  (233  KB)] for subscribers only.