Generalized Dolbeault sequences in parabolic geometry

Abstract

In this paper, 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 can be locally identified with the Dirac operator in $k$ Clifford variables, $D=(D_1,...,D_k)$, where $D_i={\sum }_j{e}_j\cdot {\partial }_{ij}:C^{\infty }(({\mathbb{R}}^n{)}^k,\S )\to C^{\infty }(({\mathbb{R}}^n{)}^k,\S )$. 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, what are purely algebraic objects.

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• $k$-Dirac Complexes