Generalized Dolbeault sequences in parabolic geometry

TL;DR

This paper constructs invariant differential operator sequences on a homogeneous model of parabolic geometry, generalizing Dirac operators, with detailed structure in odd dimensions, using algebraic methods involving Verma modules.

Contribution

It introduces a new sequence of invariant differential operators in parabolic geometry, generalizing Dirac operators, and describes their structure algebraically.

Findings

Existence of invariant differential operator sequences on $G/P$

First operator identified with Dirac operator in Clifford variables

Explicit structure described for odd-dimensional cases

Abstract

In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model $G/P$ of a Cartan geometry. The first operator in this sequence can be locally identified with the Dirac operator in $k$ Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\S)\to C^\infty((\R^n)^k,\S)$. We describe the structure of these sequences in case the dimension $n$ is odd. It follows from the construction that all these operators are invariant with respect to the action of the group $G$. These results are obtained by constructing homomorphisms of generalized Verma modules, what are purely algebraic objects.