Journal of Lie Theory

no. 4
Generalized Dolbeault Sequences in Parabolic Geometry
Journal of Lie Theory, Volume 18 (2008) no. 4, pp. 757-773
\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.
DOI: 10.5802/jolt.524
Classification: 58J10, 34L40
Keywords: Dirac operator, parabolic geometry, BGG, generalized Verma module 
@article{JOLT_2008_18_4_a1,
     author = {P. Franek},
     title = {Generalized {Dolbeault} {Sequences} in {Parabolic} {Geometry}},
     journal = {Journal of Lie Theory},
     pages = {757--773},
     year = {2008},
     volume = {18},
     number = {4},
     doi = {10.5802/jolt.524},
     zbl = {1176.17003},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.524/}
}
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P. Franek. Generalized Dolbeault Sequences in Parabolic Geometry. Journal of Lie Theory, Volume 18 (2008) no. 4, pp. 757-773. doi: 10.5802/jolt.524

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