Reductive homogeneous spaces associated with real forms. A gauge-theoretical generalisation

Abstract: Let $G$ be a connected complex Lie group. A real form of $G$ is a closed subgroup $H\subset G$ whose Lie algebra $\mathfrak{h}$ is a real form of the Lie algebra $\mathfrak{g}$ of $G$. A pair $(G,H)$ of this type is reductive, and the corresponding quotient $G/H$ is a reductive homogeneous space whose canonical connection is torsion free. Regarded as a principal $H$-bundle over $G/H$, $G$ comes with tensorial 1-form $\alpha$ of type $\mathrm{Ad}$ and a natural left invariant connection $A$. This remark suggests the following gauge theoretical generalisation of the class of reductive pairs of the form $(G,H)$ as above: Let $H$ be an arbitrary Lie group. A triple $(P\stackrel{\pi}{\to}M,\alpha,A)$, where $P\stackrel{\pi}{\to}M$ is a principal $H$-bundle, $\alpha$ a tensorial 1-form of type $\mathrm{Ad}$ on $P$ and $A$ a connection on $P$ will be called admissible if the induced linear maps $A_y\to \mathfrak{h}$, $y\in P$, are all isomorphisms. If this is the case one obtains a canonical linear connection $\nabla^\alpha_A$ on $M$ and a canonical almost complex structure $J^\alpha_A$ on $P$ which, by a result of R. Zentner, is integrable if an only if the pair $(\alpha,A)$ satisfies a gauge invariant first order differential system. A triple $(P\stackrel{\pi}{\to}M,\alpha,A)$ as above will be called integrable if this integrability condition is satisfied. Any integrable triple $(P\stackrel{\pi}{\to}M,\alpha,A)$ with $M$ simply connected and $\nabla^\alpha_A$ complete can be identified with the triple associated with a real form of a complex Lie group. In this article we explain the strategy of the proof of this classification result and we prove in detail a theorem which plays an important role in this strategy and is of independent interest. In the last section we introduce the moduli spaces of integrable pairs on a principal bundle, and we give explicit examples.