Singular BGG complexes over isotropic 2-Grassmannian

TL;DR

This paper constructs exact sequences of invariant differential operators on sections of homogeneous vector bundles over the isotropic 2-Grassmannian, extending BGG complexes to singular infinitesimal characters using Penrose transform techniques.

Contribution

It introduces singular BGG complexes over the isotropic 2-Grassmannian, providing new resolutions in a non-Hermitian setting via a novel application of the Penrose transform.

Findings

Constructed exact sequences of invariant differential operators.

Extended BGG resolutions to singular infinitesimal characters.

Applied Penrose transform in a new geometric context.

Abstract

We construct exact sequences of invariant differential operators acting on sections of certain homogeneous vector bundles in singular infinitesimal character, over the isotropic $2$-Grassmannian. This space is equal to $G/P$, where $G$ is $\operatorname{Sp}(2n,\mathbb{C})$, and $P$ its standard parabolic subgroup having the Levi factor $\operatorname{GL}(2,\mathbb{C}) \times \operatorname{Sp}(2n-4,\mathbb{C})$. The constructed sequences are analogues of the Bernstein-Gelfand-Gelfand resolutions. We do this by considering the Penrose transform over an appropriate double fibration. The result differs from the Hermitian situation.