k-Dirac Operator and Parabolic Geometries

Abstract

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the principal group of a particular type of parabolic geometry. These operators form sequences which are related to the minimal resolutions of the k-Dirac operators studied in Clifford analysis.

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

“…We have V 0 ∼ = S and V 1 ∼ = C k ⊗ S as vector spaces and we view V 1 as the vector space of k-tuples of spinors as in (1.1). It is shown in [25] that where f ∈ C ∞ (G − , S) and ε α . ∈ End(S) is the usual action of ε α on S.…”

Penrose transform and monogenic sections

“…This space is the homogeneous model for a parabolic geometry. For each fc > 2 and n > 2fc there is a sequence of invariant differential operators starting with a first order operator called the k-Dirac operator (in the parabolic setting), see [8]. These sequences belong to singular character and are interesting from the point of the fc-Dirac operator (in the Euclidean setting) studied in Clifford analysis, see [4].…”

“…In this paper we study the real analytic monogenic sections of the 2-Dirac operator Di in the case n = 6 over an afiine subset of VK (2,8). In this case we can write down explicitly the isomorphism (12) given by the Penrose transform between the third sheaf cohomology group H^{W,Ox) and the kernel of Di over the affine subset.…”

“…If A e M{n + fc,fc,C) is a matrix, then we associate to A the fc-plane in C"+'^ which is spanned by the columns of the matrix A. Xo}. The space G/P is naturally isomorphic to the Grassmannian Vc (2,8). Let Go be the stabilizer of xg := (63,64,65,63,64,65) in P. Then Go is a maximal reductive subgroup of P isomorphic to GL(2,C) x S0(6, C).…”

“…The map exp is a biholomorphism between 0_ and G_ and the map TT is a biholomorphism between G_ and U. Thus U is an affine subset of Vc (2,8) biholomorphic to M(6,2,C) x .4(2, C). …”