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Kaehler structures on $$K_ \mathbb{C}/N$$. (English) Zbl 0809.53060
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, 181-195 (1993).
Let $$K$$ be a compact semi-simple Lie group, $$G = K_ \mathbb{C}$$ and let $$KAN$$ be the Iwasawa decomposition of $$G$$. Furthermore, let $$T$$ be the centralizer of $$A$$ in $$K$$ and $$X = G/N$$. The authors give the complete classification of $$K \times T$$- invariant Kähler structures on $$X$$ being defined by a potential function $$F$$, i.e. of the form $$\omega_ F = \sqrt {-1} \partial \overline{\partial} F$$, where $$F$$ is $$K$$-invariant. The two form $$\omega_ F$$ is a Kähler form, as the authors prove if and only if $$F$$ is a strictly convex function, and the image of the mapping $${1\over 2} {\partial F\over \partial x} : \mathbb{R}^ n \to \mathbb{R}^{n_ *} = {\mathbf t}^*$$ is entirely contained in $$\text{Int }{\mathbf t}^*_ +$$. Moreover, the image of the moment map $$J : X \to {\mathbf t}^*$$, is contained in the interior of the positive Weyl chamber. Finally, if $$\lambda$$ is an integer lattice point in $${\mathbf t}^*_ +$$, then the irreducible representation of $$K$$ with maximal weight $$\lambda$$, occurs as a subrepresentation of $$\rho_ F$$ if and only if $$\lambda$$ is in the image of the moment map $$J$$. The authors study also $$\omega_ F$$ for some special choices of $$F$$. Let us mention the choice $$F(x) = \sum e^{\lambda(x)}$$, where $$\lambda$$ sums over a set of fundamental dominant weights of $$K$$.
For the entire collection see [Zbl 0780.00026].