$k$-Dirac Complexes

Abstract

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

References in corpus (7)

• Relative BGG sequences; II. BGG machinery and invariant operators
• Parabolic conformally symplectic structures II; parabolic contactification
• Relative BGG sequences; I. Algebra
• BGG complexes in singular infinitesimal character for type A
• Resolution of the $k$-Dirac operator
• Parabolic conformally symplectic structures III; Invariant differential operators and complexes
• k-Dirac operator and the Cartan-Kahler theorem for weighted differential operators

Cited by in corpus (1)

• Elliptic complex on the Grassmannian of oriented 2-planes