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This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and algebraic points of view. The notes focus on comparison theorems between the algebraic, analytic, and continuous categories.Contents include: 1.1 sheaf theory, ringed spaces; 1.2 local structure of analytic and algebraic sets; 1.3 Pn 2.1 sheaves of modules; 2.2 vector bundles; 2.3 sheaf cohomology and computations on Pn; 3.1 maximum principle and Schwarz lemma on analytic spaces; 3.2 Siegel's theorem; 3.3 Chow's theorem; 4.1 GAGA; 5.1 line bundles, divisors, and maps to Pn; 5.2 Grassmanians and vector bundles; 5.3 Chern classes and curvature; 5.4 analytic cocycles; 6.1 K-theory and Bott periodicity; 6.2 K-theory as a generalized cohomology theory; 7.1 the Chern character and obstruction theory; 7.2 the Atiyah-Hirzebruch spectral sequence; 7.3 K-theory on algebraic varieties; 8.1 Stein manifold theory; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding remarks; bibliography.Originally published in 1974.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Geometry, Algebraic --- Geometry, Analytic --- Additive map. --- Affine variety. --- Algebraic curve. --- Algebraic geometry. --- Algebraic operation. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic set. --- Analytic space. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Bott periodicity theorem. --- Cauchy's integral formula. --- Chern class. --- Chow's theorem. --- Codimension. --- Coherence condition. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Comparison theorem. --- Complex manifold. --- Complex vector bundle. --- De Rham cohomology. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential geometry. --- Differential topology. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Duality (mathematics). --- Epimorphism. --- Equation. --- Exact sequence. --- Fiber bundle. --- Fundamental class. --- General linear group. --- Grassmannian. --- Group homomorphism. --- Hilbert's syzygy theorem. --- Holomorphic function. --- Holomorphic sheaf. --- Holomorphic vector bundle. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian matrix and determinant. --- K-theory. --- Laurent series. --- Lie derivative. --- Line bundle. --- Linear map. --- Manifold. --- Mathematical induction. --- Measure (mathematics). --- Meromorphic function. --- Module (mathematics). --- Morphism. --- Neighbourhood (mathematics). --- Obstruction theory. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Principal bundle. --- Product topology. --- Projective space. --- Projective variety. --- Resolution of singularities. --- Riemann surface. --- Ring (mathematics). --- Ring homomorphism. --- Schwarz lemma. --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of modules. --- Simplicial complex. --- Special case. --- Spectral sequence. --- Stein manifold. --- Submanifold. --- Subset. --- Surjective function. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Topology. --- Transcendence degree. --- Variable (mathematics). --- Vector bundle. --- Weierstrass preparation theorem. --- Zariski topology.

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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.