Generalized Dolbeault Sequences in Parabolic Geometry

P. Franek

Journal of Lie theory, Tome 18 (2008) no. 4, pp. 757-773.

Résumé

\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.

Détail

Classification : 58J10, 34L40
Mots-clés : Dirac operator, parabolic geometry, BGG, generalized Verma module