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The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.Originally published in 1990.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.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.