Cycle Spaces of Flag Domains - (Progress in Mathematics) by Gregor Fels & Alan Huckleberry & Joseph a Wolf (Hardcover)
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<p/><br></br><p><b> Book Synopsis </b></p></br></br>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 situatio<p/><br></br><p><b> From the Back Cover </b></p></br></br><p>This monograph, divided into four parts, presents a comprehensive treatment and systematic examination of cycle spaces of flag domains. Assuming only a basic familiarity with the concepts of Lie theory and geometry, this work presents a complete structure theory for these cycle spaces, as well as their applications to harmonic analysis and algebraic geometry.</p> <p></p> <p>Key features: </p> <p>* Accessible to readers from a wide range of fields, with all the necessary background material provided for the nonspecialist</p> <p>* Many new results presented for the first time</p> <p>* Driven by numerous examples</p> <p>* The exposition is presented from the complex geometric viewpoint, but the methods, applications and much of the motivation also come from real and complex algebraic groups and their representations, as well as other areas of geometry</p> <p>* Comparisons with classical Barlet cycle spaces are given</p> <p>* Good bibliography and index</p> <p></p> <p>Researchers and graduate students in differential geometry, complex analysis, harmonic analysis, representation theory, transformation groups, algebraic geometry, and areas of global geometric analysis will benefit from this work.</p><p/><br></br><p><b> Review Quotes </b></p></br></br><br><p>From the reviews: </p> <p>"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)</p><p>"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)</p><br>
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