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Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on "ients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

Mumford-Tate groups. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Hodge theory --- Mumford-Tate groups --- Deligne torus integer. --- Galois group. --- Grothendieck conjecture. --- Hodge decomposition. --- Hodge domain. --- Hodge filtration. --- Hodge orientation. --- Hodge representation. --- Hodge structure. --- Hodge tensor. --- Hodge theory. --- Kubota rank. --- Lie algebra representation. --- Lie group. --- Mumford-Tate domain. --- Mumford-Tate group. --- Mumford-Tate subdomain. --- Noether-Lefschetz locus. --- Vogan diagram method. --- Weyl group. --- abelian variety. --- absolute Hodge class. --- algebraic geometry. --- arithmetic group. --- automorphic cohomology. --- classical group. --- compact dual. --- complex manifold. --- complex multiplication Hodge structure. --- complex multiplication. --- endomorphism algebra. --- exceptional group. --- holomorphic mapping. --- homogeneous complex manifold. --- homomorphism. --- mixed Hodge structure. --- moduli space. --- monodromy group. --- natural symmetry group. --- oriented imaginary number fields. --- period domain. --- period map. --- polarization. --- polarized Hodge structure. --- pure Hodge structure. --- reflex field. --- semisimple Lie algebra. --- semisimple Lie group. --- Γ-equivalence classes.

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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.

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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This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.

512.73 --- Harmonic analysis --- Homology theory --- Representations of groups --- Semisimple Lie groups --- Semi-simple Lie groups --- Lie groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Lie algebras. --- Lie, Algèbres de. --- Semisimple Lie groups. --- Representations of groups. --- Homology theory. --- Harmonic analysis. --- Représentations d'algèbres de Lie --- Representations of Lie algebras --- Abelian category. --- Additive identity. --- Adjoint representation. --- Algebra homomorphism. --- Associative algebra. --- Associative property. --- Automorphic form. --- Automorphism. --- Banach space. --- Basis (linear algebra). --- Bilinear form. --- Cartan pair. --- Cartan subalgebra. --- Cartan subgroup. --- Cayley transform. --- Character theory. --- Classification theorem. --- Cohomology. --- Commutative property. --- Complexification (Lie group). --- Composition series. --- Conjugacy class. --- Conjugate transpose. --- Diagram (category theory). --- Dimension (vector space). --- Dirac delta function. --- Discrete series representation. --- Dolbeault cohomology. --- Eigenvalues and eigenvectors. --- Explicit formulae (L-function). --- Fubini's theorem. --- Functor. --- Gregg Zuckerman. --- Grothendieck group. --- Grothendieck spectral sequence. --- Haar measure. --- Hecke algebra. --- Hermite polynomials. --- Hermitian matrix. --- Hilbert space. --- Hilbert's basis theorem. --- Holomorphic function. --- Hopf algebra. --- Identity component. --- Induced representation. --- Infinitesimal character. --- Inner product space. --- Invariant subspace. --- Invariant theory. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- Langlands classification. --- Langlands decomposition. --- Lexicographical order. --- Lie algebra. --- Linear extension. --- Linear independence. --- Mathematical induction. --- Matrix group. --- Module (mathematics). --- Monomial. --- Noetherian. --- Orthogonal transformation. --- Parabolic induction. --- Penrose transform. --- Projection (linear algebra). --- Reductive group. --- Representation theory. --- Semidirect product. --- Semisimple Lie algebra. --- Sesquilinear form. --- Sheaf cohomology. --- Skew-symmetric matrix. --- Special case. --- Spectral sequence. --- Stein manifold. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Summation. --- Symmetric algebra. --- Symmetric space. --- Symmetrization. --- Tensor product. --- Theorem. --- Uniqueness theorem. --- Unitary group. --- Unitary operator. --- Unitary representation. --- Upper and lower bounds. --- Verma module. --- Weight (representation theory). --- Weyl character formula. --- Weyl group. --- Weyl's theorem. --- Zorn's lemma. --- Zuckerman functor.