Generalized Complex and Dirac Structures on Homogeneous Spaces

TL;DR

This paper classifies equivariant Dirac and generalized complex structures on homogeneous spaces using Lie algebra data, covering real semisimple adjoint and nilpotent orbits, with complete results for symmetric spaces and certain compact quotients.

Contribution

It provides a Lie algebra-based description of equivariant generalized complex structures and classifies these structures on various classes of homogeneous spaces.

Findings

Complete classification for Riemannian symmetric spaces.

Classification of structures on real semisimple and nilpotent orbits.

Equivalent Lie algebra data characterizing the structures.

Abstract

We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\mathbb R$ and real nilpotent orbits in $sl_n (\mathbb R)$. We give a complete classification for Riemannian symmetric spaces and for a compact group modulo a closed, connected subgroup containing a Cartan subgroup.