Hodge theory, complex geometry, and representation theory
 Author/Creator
 Green, M. (Mark)
 Language
 English.
 Publication
 Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2013]
 Physical description
 iv, 308 pages : illustrations ; 25 cm.
 Series
 Regional conference series in mathematics ; no. 118.
Access
Available online

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QA1 .R33 NO.118
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Contributors
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 299302) and index.
 Contents

 The classical theory (2 parts)
 Polarized Hodge structures and MumfordTate groups and domains
 Hodge representations and Hodge domains
 Discrete series and ncohomology
 Geometry of flag domains (2 parts)
 Penrose transforms in the two main examples
 Automorphic cohomology
 Miscellaneous topics and some open questions.
 Publisher's Summary
 This monograph presents topics in Hodge theory and representation theory, two of the most active and important areas in contemporary mathematics. The underlying theme is the use of complex geometry to understand the two subjects and their relationships to one anotheran approach that is complementary to what is in the literature. Finitedimensional representation theory and complex geometry enter via the concept of Hodge representations and Hodge domains. Infinitedimensional representation theory, specifically the discrete series and their limits, enters through the realisation of these representations through complex geometry as pioneered by Schmid, and in the subsequent description of automorphic cohomology. For the latter topic, of particular importance is the recent work of Carayol that potentially introduces a new perspective in arithmetic automorphic representation theory. The present work gives a treatment of Carayol's work, and some extensions of it, set in a general complex geometric framework. Additional subjects include a description of the relationship between limiting mixed Hodge structures and the boundary orbit structure of Hodge domains, a general treatment of the correspondence spaces that are used to construct Penrose transforms and selected other topics from the recent literature.
(source: Nielsen Book Data)
Subjects
 Subject
 Hodge theory.
 Geometry, Differential.
 Algebraic geometry  Special varieties  Grassmannians, Schubert varieties, flag manifolds.
 Nonassociative rings and algebras  Lie algebras and Lie superalgebras  Cohomology of Lie (super)algebras.
 Topological groups, Lie groups  Locally compact groups and their algebras  Unitary representations of locally compact groups.
 Several complex variables and analytic spaces  Deformations of analytic structures  Period matrices, variation of Hodge structure; degenerations.
 Several complex variables and analytic spaces  Complex spaces with a group of automorphisms  Homogeneous complex manifolds.
 Algebraic geometry  Families, fibrations  Variation of Hodge structures.
 Algebraic geometry  Special varieties  Homogeneous spaces and generalizations.
 Nonassociative rings and algebras  Lie algebras and Lie superalgebras  Lie algebras of linear algebraic groups.
 Group theory and generalizations  Linear algebraic groups and related topics  None of the above, but in this section.
 Topological groups, Lie groups  Lie groups  Representations of Lie and linear algebraic groups over real fields: analytic methods.
 Topological groups, Lie groups  Lie groups  Semisimple Lie groups and their representations.
 Topological groups, Lie groups  Noncompact transformation groups  Homogeneous spaces.
 Several complex variables and analytic spaces  Automorphic functions  Automorphic forms.
 Several complex variables and analytic spaces  Holomorphic fiber spaces  Twistor theory, double fibrations.
 Several complex variables and analytic spaces  Complex manifolds  Stein manifolds.
 Differential geometry  Global differential geometry  Homogeneous manifolds.
Bibliographic information
 Publication date
 2013
 Responsibility
 Mark Green, Phillip Griffiths, Matt Kerr.
 Series
 Regional conference series in mathematics / Conference Board of the Mathematical Sciences ; number 118
 Note
 "Support from the National Science Foundation."
 "NSFCBMS Regional Conference in the Mathematical Sciences on Hodge Theory, Complex Geometry, and Representation Theory, held at Texas Christian University, June 1822, 2012."
 ISBN
 9781470410124 (alk. paper)
 1470410125 (alk. paper)