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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
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