Resolution of the -Dirac operator
arXiv:1705.10168 · doi:10.1007/s00006-018-0830-6
Abstract
This is the second part in a series of two papers. The -Dirac complex is a complex of differential operators which are natural to a particular -graded parabolic geometry. In this paper we will consider the -Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove that the -Dirac complex is exact with formal power series at any fixed point. Then we will show that the -Dirac complex descends from an affine subset of the homogeneous space to a complex of linear, constant coefficient differential operators and that the first operator in the descended complex is the -Dirac operator studied in Clifford analysis. The main result of this paper is that the descended complex is locally exact and thus it forms a resolution of the -Dirac operator.