# zbMATH — the first resource for mathematics

Twistor theory for indefinite Kähler symmetric spaces. (English) Zbl 0798.53066
Eastwood, Michael (ed.) et al., The Penrose transform and analytic cohomology in representation theory. AMS-IMS-SIAM summer research conference, June 27 - July 3, 1992, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 154, 117-132 (1993).
Let $$G$$ be a real semisimple Lie group and denote by $$D = G/H$$ an open orbit in a generalized flag variety for $$G^{\mathbb{C}}$$. Similar to the case of the Minkowski space and the projective space the authors discuss the Penrose transform for $$D$$: $\begin{matrix} & & G/(K \cap H)\\ & \swarrow & & \searrow \\ D & & & & G/K \end{matrix}$ If the projections are holomorphic the cohomology $$H^*(D,{\mathcal O}(\nu))$$ is related to the kernel of an invariant differential operator on $$G/K$$. The paper contains also remarks and examples in case that the projections are not holomorphic and relates the construction to conformal structures and Frobenius structures.
For the entire collection see [Zbl 0780.00026].

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 22E46 Semisimple Lie groups and their representations 53C10 $$G$$-structures