# Resolution of the $k$-Dirac operator

## TL;DR

This paper proves that the $k$-Dirac complex on a homogeneous space forms a resolution of the $k$-Dirac operator by establishing local exactness and descent properties, advancing understanding in parabolic geometry and Clifford analysis.

## Contribution

It demonstrates the local exactness of the $k$-Dirac complex and its descent to a constant coefficient differential operator complex, providing a resolution of the $k$-Dirac operator.

## Key findings

- The $k$-Dirac complex is exact with formal power series at any fixed point.
- The complex descends to a linear, constant coefficient differential operator complex.
- The descended complex is locally exact, resolving 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.