k-Dirac Complexes

Abstract

This is the first paper in a series of two papers. In this paper, we construct complexes of invariant differential operators that 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.