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Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.

Differential geometry. Global analysis --- Riemannian manifolds --- Symmetric spaces --- Rigidity (Geometry) --- 512 --- Lie groups --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Spaces, Symmetric --- Geometry, Differential --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics) --- Groups, Lie --- Lie algebras --- Topological groups --- Algebra --- Lie groups. --- Riemannian manifolds. --- Symmetric spaces. --- Rigidity (Geometry). --- 512 Algebra --- Addition. --- Adjoint representation. --- Affine space. --- Approximation. --- Automorphism. --- Axiom. --- Big O notation. --- Boundary value problem. --- Cohomology. --- Compact Riemann surface. --- Compact space. --- Conjecture. --- Constant curvature. --- Corollary. --- Counterexample. --- Covering group. --- Covering space. --- Curvature. --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Dimension. --- Direct product. --- Division algebra. --- Ergodicity. --- Erlangen program. --- Existence theorem. --- Exponential function. --- Finitely generated group. --- Fundamental domain. --- Fundamental group. --- Geometry. --- Half-space (geometry). --- Hausdorff distance. --- Hermitian matrix. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- Identity matrix. --- Inner automorphism. --- Isometry group. --- Jordan algebra. --- Matrix multiplication. --- Metric space. --- Morphism. --- Möbius transformation. --- Normal subgroup. --- Normalizing constant. --- Partially ordered set. --- Permutation. --- Projective space. --- Riemann surface. --- Riemannian geometry. --- Sectional curvature. --- Self-adjoint. --- Set function. --- Smoothness. --- Stereographic projection. --- Subgroup. --- Subset. --- Summation. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topology. --- Tubular neighborhood. --- Two-dimensional space. --- Unit sphere. --- Vector group. --- Weyl group. --- Riemann, Variétés de --- Lie, Groupes de --- Geometrie differentielle globale --- Varietes riemanniennes

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété

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This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary representations of real reductive groups, providing fairly complete definitions and references, and sketches (at least) of most proofs. The first half of the book is devoted to the three more or less understood constructions of such representations: parabolic induction, complementary series, and cohomological parabolic induction. This culminates in the description of all irreducible unitary representation of the general linear groups. For other groups, one expects to need a new construction, giving "unipotent representations." The latter half of the book explains the evidence for that expectation and suggests a partial definition of unipotent representations.

Lie groups --- Representations of Lie groups --- Lie groups. --- Representations of Lie groups. --- 512.81 --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- 512.81 Lie groups --- Abelian group. --- Adjoint representation. --- Annihilator (ring theory). --- Atiyah–Singer index theorem. --- Automorphic form. --- Automorphism. --- Cartan subgroup. --- Circle group. --- Class function (algebra). --- Classification theorem. --- Cohomology. --- Commutator subgroup. --- Complete metric space. --- Complex manifold. --- Conjugacy class. --- Cotangent space. --- Dimension (vector space). --- Discrete series representation. --- Dixmier conjecture. --- Dolbeault cohomology. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior algebra. --- Function space. --- Group homomorphism. --- Harmonic analysis. --- Hecke algebra. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Homogeneous space. --- Homomorphism. --- Induced representation. --- Infinitesimal character. --- Inner automorphism. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Isometry group. --- Isometry. --- K-finite. --- Kazhdan–Lusztig polynomial. --- Langlands decomposition. --- Lie algebra cohomology. --- Lie algebra representation. --- Lie algebra. --- Lie group action. --- Lie group. --- Mathematical induction. --- Maximal compact subgroup. --- Measure (mathematics). --- Minkowski space. --- Nilpotent group. --- Orbit method. --- Orthogonal group. --- Parabolic induction. --- Principal homogeneous space. --- Principal series representation. --- Projective space. --- Pseudo-Riemannian manifold. --- Pullback (category theory). --- Ramanujan–Petersson conjecture. --- Reductive group. --- Regularity theorem. --- Representation of a Lie group. --- Representation theorem. --- Representation theory. --- Riemann sphere. --- Riemannian manifold. --- Schwartz space. --- Semisimple Lie algebra. --- Sheaf (mathematics). --- Sign (mathematics). --- Special case. --- Spectral theory. --- Sub"ient. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic vector space. --- Tangent space. --- Tautological bundle. --- Theorem. --- Topological group. --- Topological space. --- Trivial representation. --- Unitary group. --- Unitary matrix. --- Unitary representation. --- Universal enveloping algebra. --- Vector bundle. --- Weyl algebra. --- Weyl character formula. --- Weyl group. --- Zariski's main theorem. --- Zonal spherical function. --- Représentations de groupes de Lie --- Groupes de lie --- Representation des groupes de lie

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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)