# Elliptic complex on the Grassmannian of oriented 2-planes

## TL;DR

This paper constructs an elliptic complex of invariant differential operators on the Grassmannian of oriented 2-planes, revealing its index is zero, and explores its geometric and analytical properties.

## Contribution

It introduces a new elliptic complex of invariant differential operators of length 3 on the Grassmannian of oriented 2-planes, starting with the 2-Dirac operator.

## Key findings

- The complex is elliptic and invariant under the Grassmannian's symmetry group.
- The index of the complex is zero.
- The complex begins with the 2-Dirac operator.

## Abstract

The Grassmannian $V_2(\mathbb{R}^{n+2})$ of oriented 2-planes in $\mathbb R^{n+2}$ where $n\ge3$ carries a homogeneous parabolic contact structure of Grassmannian type. The main result of this article is that on $V_2(\mathbb{R}^{n+2})$ lives an elliptic complex of invariant differential operators of length 3 which starts with the 2-Dirac operator and that the index of the complex is zero.