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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. --- Lie, Algèbres de. --- Représentations d'algèbres de Lie

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The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

Harmonic analysis. --- Harmonic functions. --- Functions, Harmonic --- Laplace's equations --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Harmonic analysis. Fourier analysis --- Harmonic analysis --- Fourier analysis --- Harmonic functions --- Analyse harmonique --- Analyse de Fourier --- Fonctions harmoniques --- Fourier Analysis --- Fourier, Transformations de --- Euclide, Espaces d' --- Bessel functions --- Differential equations, Partial --- Fourier series --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Banach algebras --- Time-series analysis --- Analysis, Fourier --- Fourier analysis. --- Basic Sciences. Mathematics --- Analysis, Functions --- Analysis, Functions. --- Calculus --- Mathematical analysis --- Mathematics --- Fourier, Transformations de. --- Euclide, Espaces d'. --- Potentiel, Théorie du --- Fonctions harmoniques. --- Potential theory (Mathematics) --- Analytic continuation. --- Analytic function. --- Banach algebra. --- Banach space. --- Bessel function. --- Borel measure. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Cartesian coordinate system. --- Cauchy–Riemann equations. --- Change of variables. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Complex plane. --- Conformal map. --- Conjugate transpose. --- Continuous function (set theory). --- Continuous function. --- Convolution. --- Differentiation of integrals. --- Dimensional analysis. --- Dirichlet problem. --- Disk (mathematics). --- Distribution (mathematics). --- Equation. --- Euclidean space. --- Existential quantification. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fubini's theorem. --- Function (mathematics). --- Function space. --- Green's theorem. --- Hardy's inequality. --- Hardy–Littlewood maximal function. --- Harmonic function. --- Hermitian matrix. --- Hilbert transform. --- Holomorphic function. --- Homogeneous function. --- Inequality (mathematics). --- Infimum and supremum. --- Interpolation theorem. --- Interval (mathematics). --- Lebesgue integration. --- Lebesgue measure. --- Linear interpolation. --- Linear map. --- Linear space (geometry). --- Line–line intersection. --- Liouville's theorem (Hamiltonian). --- Lipschitz continuity. --- Locally integrable function. --- Lp space. --- Majorization. --- Marcinkiewicz interpolation theorem. --- Mean value theorem. --- Measure (mathematics). --- Mellin transform. --- Monotonic function. --- Multiplication operator. --- Norm (mathematics). --- Operator norm. --- Orthogonal group. --- Paley–Wiener theorem. --- Partial derivative. --- Partial differential equation. --- Plancherel theorem. --- Pointwise convergence. --- Poisson kernel. --- Poisson summation formula. --- Polynomial. --- Principal value. --- Quadratic form. --- Radial function. --- Radon–Nikodym theorem. --- Representation theorem. --- Riesz transform. --- Scientific notation. --- Series expansion. --- Singular integral. --- Special case. --- Subharmonic function. --- Support (mathematics). --- Theorem. --- Topology. --- Total variation. --- Trigonometric polynomial. --- Trigonometric series. --- Two-dimensional space. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Fourier, Analyse de --- Potentiel, Théorie du. --- Potentiel, Théorie du --- Espaces de hardy