Resolution of the $${\varvec{k}}$$ k -Dirac operator

Abstract

Abstract. This is the second part in a series of two papers. The kDirac complex is a complex of differential operators which are naturally associated to a particular |2|-graded parabolic geometry. In this paper we will consider the k-Dirac complex over the homogeneous space of the parabolic geometry and as a first result, we will prove that the k-Dirac complex is formally exact (in the sense of formal power series). Then we will show that the k-Dirac complex descends from an affine subset of the homogeneous space to a complex of linear and 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.

Singular BGG complexes over isotropic 2-Grassmannian

“…Also, BGG complexes are closely related to the Cousin complexes, as in [Mil]. For other similar results in a higher grading, see [KS06], [Sal18a], [Sal18b].…”

k-Dirac Complexes

“…With such a structure, it is more natural to work with weighted jets (see [18]) rather than usual jets and we use this concept also here, i.e., we prove the formal exactness of the k-Dirac complexes with respect to the weighted jets. Nevertheless, we will prove in [24] that the formal exactness of the k-Dirac complex (or more precisely the exactness of (7.16) for each + j ≥ 0) is enough to conclude that it descends to a resolution of the k-Dirac operator.…”

“…This result is crucial for an application in [24] where it is shown that the complex is exact with formal power series at any fixed point and that it descends (as outlined in the recent series [5,6,7]) to a resolution of the k-Dirac operator studied in Clifford analysis (see [12,20]). As a potential application of the resolution, there is an open problem of characterizing the domains of monogenicity, i.e., an open set U is a domain of monogenicity if there is no open set U with U U such that each monogenic function 1 in U extends to a monogenic function in U .…”

“…For the application in [24], we need to show that the k-Dirac complexes constructed in this paper give rise to complexes from [22] which live on the corresponding real parabolic geometries. This turns out to be rather easy since any linear G-invariant operator is determined by a certain P-equivariant homomorphism.…”

“…The exactness of the complex (7.15) for each + j ≥ 0 implies (see [24]) the exactness of the k-Dirac complex at the level of infinite weighted jets at any fixed point. Following [25], we say that the k-Dirac complex is formally exact.…”