By Gregor Fels

This monograph, divided into 4 elements, offers a entire remedy and systematic exam of cycle areas of flag domain names. Assuming just a easy familiarity with the strategies of Lie idea and geometry, this paintings offers a whole constitution conception for those cycle areas, in addition to their purposes to harmonic research and algebraic geometry. Key positive aspects: * available to readers from quite a lot of fields, with all of the priceless historical past fabric supplied for the nonspecialist * Many new effects provided for the 1st time * pushed by way of quite a few examples * The exposition is gifted from the complicated geometric point of view, however the equipment, purposes and lots more and plenty of the incentive additionally come from actual and intricate algebraic teams and their representations, in addition to different components of geometry * Comparisons with classical Barlet cycle areas are given * sturdy bibliography and index. Researchers and graduate scholars in differential geometry, complicated research, harmonic research, illustration thought, transformation teams, algebraic geometry, and components of world geometric research will make the most of this paintings.

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**Additional info for Cycle Spaces of Flag Domains. A Complex Geometric Viewpoint**

**Example text**

Those concepts later translate into cohomology vanishing theorems for holomorphic vector bundles over ﬂag domains. In Section 7 we see that measurable open orbits carry certain canonical exhaustion functions, and that those exhaustion functions give a measure of the holomorphic convexity/concavity of the open orbit, thus making the cohomology vanishing theorems explicit for the case of measurable ﬂag domains. 1 Automorphisms and regular elements Fix a Cartan involution θ of g0 and G0 . It is an automorphism of square 1, and K0 = Gθ0 is a maximal compact subgroup of G0 .

Without loss of generality we conjugate by b−1 . Now we may assume x = sw x0 . Then B = Ad(sw )B and therefore b = ad(sw )b, which contains h. Every h ∈ H normalizes both b and b , so h ∈ B ∩ B . Thus the intersection of two Borel subgroups contains a Cartan subgroup. The lemma follows. Let G0 be a real form of G. In other words, G0 is a Lie subgroup of G whose Lie algebra g0 is a real form of g. Although G is connected, G0 does not have to be connected. We always write τ both for the complex conjugation of g over g0 and for the corresponding conjugation of G over G0 .

We make the standard choice, + (g, h) = {εi − εj | 1 i