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Book
Non-spherical principal series representations of a semisimple Lie group
Author:
ISBN: 0821822160 Year: 1979 Publisher: Providence (R.I.): American Mathematical Society

Enright-Shelton theory and Vogan's problem for generalized principal series
Authors: ---
ISBN: 082182547X Year: 1993 Publisher: Providence (R.I.): American Mathematical Society

Conjugacy classes in semisimple algebraic groups.
Author:
ISBN: 0821803336 Year: 1995 Publisher: Providence American Mathematical Society

The action of a real semisimple Lie group on a complex flag manifold
Author:
ISBN: 0821818384 Year: 1974 Publisher: Providence (R.I.): American Mathematical Society


Book
Flag varieties : an interplay of geometry, combinatorics, and representation theory
Authors: ---
ISBN: 9811313938 9811313946 9789811313936 Year: 2018 Publisher: Singapore : Springer,

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Abstract

This book discusses the importance of flag varieties in geometric objects and elucidates its richness as interplay of geometry, combinatorics and representation theory. The book presents a discussion on the representation theory of complex semisimple Lie algebras, as well as the representation theory of semisimple algebraic groups. In addition, the book also discusses the representation theory of symmetric groups. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Many of the geometric results admit elegant combinatorial description because of the root system connections, a typical example being the description of the singular locus of a Schubert variety. This discussion is carried out as a consequence of standard monomial theory. Consequently, this book includes standard monomial theory and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. The two recent results on Schubert varieties in the Grassmannian have also been included in this book. The first result gives a free resolution of certain Schubert singularities. The second result is about certain Levi subgroup actions on Schubert varieties in the Grassmannian and derives some interesting geometric and representation-theoretic consequences.

Cycle spaces of flag domains : a complex geometric viewpoint
Authors: --- ---
ISBN: 1280611278 9786610611270 0817644792 0817643915 Year: 2006 Publisher: Boston : Birkhauser,

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Abstract

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. Key features: * Accessible to readers from a wide range of fields, with all the necessary background material provided for the nonspecialist * Many new results presented for the first time * Driven by numerous examples * 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 * Comparisons with classical Barlet cycle spaces are given * Good bibliography and index 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.

Keywords

Semisimple Lie groups. --- Flag manifolds. --- Twistor theory. --- Automorphic forms. --- Homogeneous spaces. --- Spaces, Homogeneous --- Lie groups --- Automorphic functions --- Forms (Mathematics) --- Twistors --- Congruences (Geometry) --- Field theory (Physics) --- Space and time --- Flag varieties (Mathematics) --- Manifolds, Flag --- Varieties, Flag (Mathematics) --- Algebraic varieties --- Semi-simple Lie groups --- Global differential geometry. --- Topological Groups. --- Differential equations, partial. --- Global analysis. --- Geometry, algebraic. --- Quantum theory. --- Differential Geometry. --- Topological Groups, Lie Groups. --- Several Complex Variables and Analytic Spaces. --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Quantum Physics. --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Algebraic geometry --- Geometry --- Partial differential equations --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Differential geometry. --- Topological groups. --- Lie groups. --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Algebraic geometry. --- Quantum physics. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Topology --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential geometry

Representation theory of semisimple groups: an overview based on examples
Author:
ISBN: 0691090890 9780691090894 0691084017 1400883970 9780691084015 Year: 1986 Volume: 36 Publisher: Princeton, N.J.

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Abstract

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Keywords

Semisimple Lie groups. --- Representations of groups. --- Groupes de Lie semi-simples --- Représentations de groupes --- Semisimple Lie groups --- Representations of groups --- Semi-simple Lie groups --- Lie groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Représentations de groupes --- 512.547 --- 512.547 Linear representations of abstract groups. Group characters --- Linear representations of abstract groups. Group characters --- Abelian group. --- Admissible representation. --- Algebra homomorphism. --- Analytic function. --- Analytic proof. --- Associative algebra. --- Asymptotic expansion. --- Automorphic form. --- Automorphism. --- Bounded operator. --- Bounded set (topological vector space). --- Cartan subalgebra. --- Cartan subgroup. --- Category theory. --- Characterization (mathematics). --- Classification theorem. --- Cohomology. --- Complex conjugate representation. --- Complexification (Lie group). --- Complexification. --- Conjugate transpose. --- Continuous function (set theory). --- Degenerate bilinear form. --- Diagram (category theory). --- Dimension (vector space). --- Dirac operator. --- Discrete series representation. --- Distribution (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Existence theorem. --- Explicit formulae (L-function). --- Fourier inversion theorem. --- General linear group. --- Group homomorphism. --- Haar measure. --- Heine–Borel theorem. --- Hermitian matrix. --- Hilbert space. --- Holomorphic function. --- Hyperbolic function. --- Identity (mathematics). --- Induced representation. --- Infinitesimal character. --- Integration by parts. --- Invariant subspace. --- Invertible matrix. --- Irreducible representation. --- Jacobian matrix and determinant. --- K-finite. --- Levi decomposition. --- Lie algebra. --- Locally integrable function. --- Mathematical induction. --- Matrix coefficient. --- Matrix group. --- Maximal compact subgroup. --- Meromorphic function. --- Metric space. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Parity (mathematics). --- Plancherel theorem. --- Projection (linear algebra). --- Quantifier (logic). --- Reductive group. --- Representation of a Lie group. --- Representation theory. --- Schwartz space. --- Semisimple Lie algebra. --- Set (mathematics). --- Sign (mathematics). --- Solvable Lie algebra. --- Special case. --- Special linear group. --- Special unitary group. --- Subgroup. --- Summation. --- Support (mathematics). --- Symmetric algebra. --- Symmetrization. --- Symplectic group. --- Tensor algebra. --- Tensor product. --- Theorem. --- Topological group. --- Topological space. --- Topological vector space. --- Unitary group. --- Unitary matrix. --- Unitary representation. --- Universal enveloping algebra. --- Variable (mathematics). --- Vector bundle. --- Weight (representation theory). --- Weyl character formula. --- Weyl group. --- Weyl's theorem. --- ZPP (complexity). --- Zorn's lemma.

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