Resolution of the $k$-Dirac operator

Abstract

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove that the $k$-Dirac complex is exact with formal power series at any fixed point. Then we will show that the $k$-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 $k$-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 $k$-Dirac operator.