The Hitchin–Kostant–Rallis section conjecture for real moduli spaces

Let $\mathfrak{g}_{\mathbb{R}}$ be a real form, let $\hat{\mathcal{T}}_{\mathfrak{g}_{\mathbb{R}}}$ be the moduli space of $\mathfrak{g}_{\mathbb{R}}$-complex structures, and let the Hitchin–Kostant–Rallis section be the section of the corresponding invariant-polynomial fibration. Hitchin–Kostant–Rallis section conjecture. The moduli space $\hat{\mathcal{T}}_{\mathfrak{g}_{\mathbb{R}}}$ is canonically homeomorphic to the image of the Hitchin–Kostant–Rallis section. The split-real case is related in the paper to the complex moduli space and the ordinary Hitchin section, but the asserted identification for general real forms is left conjectural.