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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].
Reviewer: N.Bokan (Beograd)

53C35 Differential geometry of symmetric spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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