Cycle Spaces of Flag Domains

1. Front Matter

   Pages i-xx

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2. Introduction to Flag Domain Theory

   1. Front Matter

      Pages 1-4

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   2. Structure of Complex Flag Manifolds

      Pages 5-26

   3. Real Group Orbits

      Pages 27-30

   4. Orbit Structure for Hermitian Symmetric Spaces

      Pages 31-36

   5. Open Orbits

      Pages 37-53

   6. The Cycle Space of a Flag Domain

      Pages 55-71

3. Cycle Spaces as Universal Domains

   1. Front Matter

      Pages 73-76

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   2. Universal Domains

      Pages 77-91

   3. B-Invariant Hypersurfaces in M_Z

      Pages 93-111

   4. Orbit Duality via Momentum Geometry

      Pages 113-124

   5. Schubert Slices in the Context of Duality

      Pages 125-132

   6. Analysis of the Boundary of U

      Pages 133-161

   7. Invariant Kobayashi-Hyperbolic Stein Domains

      Pages 163-173

   8. Cycle Spaces of Lower-Dimensional Orbits

      Pages 175-184

   9. Examples

      Pages 185-201

4. Analytic and Geometric Consequences

   1. Front Matter

      Pages 203-206

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   2. The Double Fibration Transform

      Pages 207-215

   3. Variation of Hodge Structure

      Pages 217-223

   4. Cycles in the K3 Period Domain

      Pages 225-237

This research monograph is a systematic exposition of the background, methods, and recent results in the theory of cycle spaces of ?ag domains. Some of the methods are now standard, but many are new. The exposition is carried out from the viewpoint of complex algebraic and differential geometry. Except for certain foundational material,whichisreadilyavailablefromstandardtexts,itisessentiallyself-contained; at points where this is not the case we give extensive references. After developing the background material on complex ?ag manifolds and rep- sentationtheory, wegiveanexposition(withanumberofnewresults)ofthecomplex geometric methods that lead to our characterizations of (group theoretically de?ned) cyclespacesandtoanumberofconsequences. Thenwegiveabriefindicationofjust how those results are related to the representation theory of semisimple Lie groups through, for example, the theory of double ?bration transforms, and we indicate the connection to the variation of Hodge structure. Finally, we work out detailed local descriptions of the relevant full Barlet cycle spaces. Cycle space theory is a basic chapter in complex analysis. Since the 1960s its importance has been underlined by its role in the geometry of ?ag domains, and by applications in the representation theory of semisimple Lie groups. This developed veryslowlyuntilafewofyearsagowhenmethodsofcomplexgeometry,inparticular those involving Schubert slices, Schubert domains, Iwasawa domains and suppo- ing hypersurfaces, were introduced. In the late 1990s, and continuing through early 2002, we developed those methods and used them to give a precise description of cycle spaces for ?ag domains. This effectively enabled the use of double ?bration transforms in all ?ag domain situations.

• Complex analysis
• Representation theory
• differential geometry
• harmonic analysis
• manifold

From the reviews:

"Cycle spaces can be a useful tool in the study of real semisimple Lie groups, and the research monograph which is reviewed here is devoted to describing their features. The exposition … is in principle self-contained for a good graduate reader, who will also find a wealth of concrete examples. … the approach used by the authors throughout this monograph is based on a combination of group-theoretical methods … the result is an intriguing melting pot, opening interesting perspectives of interaction among different research branches." (Corrado Marastoni, Mathematical Reviews, Issue 2006 h)

“A systematic exposition of the background, methods, and recent results in the theory of cycle spaces of flag domains. … The value of this progress in mathematics volume to a wide group of researchers … is indisputable. They all will admire the volume for the many new results presented for the first time. Your reviewer would strongly recommend that you spend a few hours with this volume long enough to familiarize yourself with its contents. You’ll be back for the details when you need them.” (Current Engineering Practice, Vol. 48, 2005-2006)

• Fakultät für Mathematik und Physik, Universität Tübingen, Tübingen, Germany

  Gregor Fels

• Institut für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

  Alan Huckleberry

• Department of Mathematics, University of California, Berkeley, USA

  Joseph A. Wolf

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