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Unitarity of certain Dolbeault cohomology representations. (English) Zbl 0842.22012
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, 239-259 (1993).
Let \(G\) be a connected linear semisimple Lie group and \(H\) its subgroup which is a centralizer of a torus. Then \(G/H\) has a natural structure of a homogeneous complex manifold. Let \(\chi\) be a unitary character of \(H\) and \({\mathcal L}_\chi\) the corresponding homogeneous line bundle. The author studies (under some additional technical conditions) the unitarizability of representations of \(G\) on Dolbeault cohomology spaces \(H^s(G/H, {\mathcal L}_\chi)\), \(s \in \mathbb{Z}\), of \({\mathcal L}_\chi\).
For the entire collection see [Zbl 0780.00026].

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E46 Semisimple Lie groups and their representations
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