# $k$-Dirac Complexes

## TL;DR

This paper introduces $k$-Dirac complexes as invariant differential operator complexes on certain homogeneous spaces, showing their construction via the Penrose transform and their formal exactness, with applications to Clifford analysis.

## Contribution

It constructs and analyzes the $k$-Dirac complexes, linking them to relative BGG sequences and the Penrose transform, and proves their formal exactness.

## Key findings

- $k$-Dirac complexes are formally exact sequences.
- They arise as direct images of relative BGG sequences.
- They provide resolutions of the $k$-Dirac operator.

## Abstract

This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.