On Flag Domains in the Supersymmetric Setting

TL;DR

This thesis explores the structure and properties of flag domains in the supersymmetric setting, focusing on measurability, holomorphic functions, and links to representation theory, providing classifications and revising classical results.

Contribution

It classifies all measurable flag domains and global holomorphic functions, and connects real reductive supergroup representations with Lie superalgebra theory.

Findings

Classified all measurable flag domains.

Provided a classification of global holomorphic functions.

Linked representation theory of supergroups with Lie superalgebra theory.

Abstract

Flag domains are open orbits of real forms $G_\mathbb{R}$ of complex reductive Lie supergroups $G$ in $G$-flag supermanifolds $Z = G/P$. This thesis discusses three topics from the theory of these flag domains: 1. Measurability(i.e. existence of $G_\mathbb{R}$-invariant Berezinian densities) 2. Global holomorphic superfunctions 3. Cycle spaces and the Double Fibration Transform A revision of the respective classical results is included in the second chapter. This thesis provides a classification of all measurable flag domains and of the global holomorphic functions on all flag domains. Moreover, the last chapter provides a possible link between the representation theory of real reductive (super) Lie groups and the Bott-Borel-Weyl Theory for complex Lie superalgebras.