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This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

Lie groups. --- Algebra. --- Analytic group. --- Approximate identity. --- Associative algebra. --- Associativity formulas. --- Borel subalgebra. --- Borel-Weil Theorem. --- Boundary operator. --- Cartan subgroup. --- Chain map. --- Change of rings. --- Closed linear group. --- Cohomological induction. --- Complexification. --- Convolution. --- Derivation. --- Derived functor. --- Diffeomorphism. --- Divisible group. --- Dual vector space. --- Enough injectives. --- Equivalence. --- Euler-Poincare principle. --- Exponential. --- Forgetful functor. --- Free resolution. --- Functor. --- Good category. --- Haar measure. --- Highest weight. --- Homogeneous polynomial. --- Homogeneous tensor. --- I functor. --- Induction, cohomological. --- Inner derivation. --- Isotypic subspace. --- K-isotypic subspace. --- Left exact functor. --- Lie algebra. --- Lie bracket. --- Linear extension. --- Long exact sequence. --- Mackey isomorphism. --- Matrix coefficient. --- Multiplicity. --- Normalized Haar measure. --- Positive root. --- Quotient representation. --- Reductive group. --- Regular representation. --- Right exact functor. --- Schur orthogonality. --- Semidirect product. --- Spectral sequence.

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The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of automorphic representations. This work represents the first case in which the stable trace formula has been worked out beyond the case of SL (2) and related groups. Many phenomena which will appear in the general case present themselves already for these unitary groups.

Unitary groups. --- Trace formulas. --- Representations of groups. --- Automorphic forms. --- Abelian group. --- Abuse of notation. --- Addition. --- Admissible representation. --- Algebraic closure. --- Algebraic group. --- Algebraic number field. --- Asymptotic expansion. --- Automorphism. --- Base change map. --- Base change. --- Bijection. --- Borel subgroup. --- Cartan subgroup. --- Class function (algebra). --- Coefficient. --- Combination. --- Compact group. --- Complementary series representation. --- Complex number. --- Congruence subgroup. --- Conjugacy class. --- Continuous function. --- Corollary. --- Countable set. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Dimension. --- Discrete spectrum. --- Division algebra. --- Division by zero. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Existential quantification. --- Finite set. --- Fourier transform. --- Fundamental lemma (Langlands program). --- G factor (psychometrics). --- Galois group. --- Global field. --- Haar measure. --- Hecke algebra. --- Homomorphism. --- Hyperbolic set. --- Index notation. --- Irreducible representation. --- Isomorphism class. --- L-function. --- Langlands classification. --- Linear combination. --- Local field. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal torus. --- Morphism. --- Multiplicative group. --- Neighbourhood (mathematics). --- Orbital integral. --- Oscillator representation. --- P-adic number. --- Parity (mathematics). --- Principal series representation. --- Quaternion algebra. --- Quaternion. --- Reductive group. --- Regular element. --- Remainder. --- Representation theory. --- Ring of integers. --- Scientific notation. --- Semisimple algebra. --- Set (mathematics). --- Shimura variety. --- Simple algebra. --- Smoothness. --- Special case. --- Stable distribution. --- Subgroup. --- Summation. --- Support (mathematics). --- Tate conjecture. --- Tensor product. --- Theorem. --- Trace formula. --- Triangular matrix. --- Unitary group. --- Variable (mathematics). --- Weight function. --- Weil group.

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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.

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Based on a series of lectures given by Harish-Chandra at the Institute for Advanced Study in 1971-1973, this book provides an introduction to the theory of harmonic analysis on reductive p-adic groups.Originally published in 1979.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.

512.74 --- p-adic groups --- Banach algebras --- Groups, p-adic --- Algebraic groups. Abelian varieties --- p-adic groups. --- 512.74 Algebraic groups. Abelian varieties --- P-adic groups. --- Harmonic analysis. Fourier analysis --- Harmonic analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Group theory --- Harmonic analysis. --- Adjoint representation. --- Admissible representation. --- Algebra homomorphism. --- Algebraic group. --- Analytic continuation. --- Analytic function. --- Associative property. --- Automorphic form. --- Automorphism. --- Banach space. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subgroup. --- Central simple algebra. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Class function (algebra). --- Commutative property. --- Compact space. --- Composition series. --- Conjugacy class. --- Corollary. --- Dimension (vector space). --- Discrete series representation. --- Division algebra. --- Double coset. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existential quantification. --- Factorization. --- Fourier series. --- Function (mathematics). --- Functional equation. --- Fundamental domain. --- Fundamental lemma (Langlands program). --- G-module. --- Group isomorphism. --- Haar measure. --- Hecke algebra. --- Holomorphic function. --- Identity element. --- Induced representation. --- Inner automorphism. --- Lebesgue measure. --- Levi decomposition. --- Lie algebra. --- Locally constant function. --- Locally integrable function. --- Mathematical induction. --- Matrix coefficient. --- Maximal compact subgroup. --- Meromorphic function. --- Module (mathematics). --- Module homomorphism. --- Open set. --- Order of integration (calculus). --- Orthogonal complement. --- P-adic number. --- Pole (complex analysis). --- Product measure. --- Projection (linear algebra). --- Quotient module. --- Quotient space (topology). --- Radon measure. --- Reductive group. --- Representation of a Lie group. --- Representation theorem. --- Representation theory. --- Ring homomorphism. --- Schwartz space. --- Semisimple algebra. --- Separable extension. --- Sesquilinear form. --- Set (mathematics). --- Sign (mathematics). --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Surjective function. --- Tempered representation. --- Tensor product. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Trace (linear algebra). --- Transitive relation. --- Unitary representation. --- Universal enveloping algebra. --- Variable (mathematics). --- Vector space. --- Analyse harmonique (mathématiques)