
Journal of Lie Theory 18 (2008), No. 4, 757774 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 [Abstractpdf] \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 [ Fulltextpdf (233 KB)] for subscribers only. 