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In this book Professor Lusztig solves an interesting problem by entirely new methods: specifically, the use of cohomology of buildings and related complexes.The book gives an explicit construction of one distinguished member, D(V), of the discrete series of GLn (Fq), where V is the n-dimensional F-vector space on which GLn(Fq) acts. This is a p-adic representation; more precisely D(V) is a free module of rank (q--1) (q2-1)...(qn-1-1) over the ring of Witt vectors WF of F. In Chapter 1 the author studies the homology of partially ordered sets, and proves some vanishing theorems for the homology of some partially ordered sets associated to geometric structures. Chapter 2 is a study of the representation △ of the affine group over a finite field. In Chapter 3 D(V) is defined, and its restriction to parabolic subgroups is determined. In Chapter 4 the author computes the character of D(V), and shows how to obtain other members of the discrete series by applying Galois automorphisms to D(V). Applications are in Chapter 5. As one of the main applications of his study the author gives a precise analysis of a Brauer lifting of the standard representation of GLn(Fq).

Group theory --- Algebraic fields --- Linear algebraic groups --- Representations of groups --- Series --- 511.33 --- Algebra --- Mathematics --- Processes, Infinite --- Sequences (Mathematics) --- Group representation (Mathematics) --- Groups, Representation theory of --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Algebraic fields. --- Linear algebraic groups. --- Representations of groups. --- Series. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Addition. --- Affine group. --- Automorphism. --- Dimension. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Field of fractions. --- Finite field. --- Free module. --- Grothendieck group. --- Homomorphism. --- Linear subspace. --- Morphism. --- P-adic number. --- Partially ordered set. --- Simplicial complex. --- Tensor product. --- Theorem. --- Witt vector. --- Groupes algébriques linéaires --- Groupes algébriques linéaires --- Représentations de groupes

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The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n+3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P=m. For n=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU(1,n). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n-variables of the Kummer identities for n-1 involving quadratic and cubic changes of the variable.

Hypergeometric functions. --- Monodromy groups. --- Lattice theory. --- Abuse of notation. --- Algebraic variety. --- Analytic continuation. --- Arithmetic group. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Codimension. --- Coefficient. --- Cohomology. --- Commensurability (mathematics). --- Compactification (mathematics). --- Complete quadrangle. --- Complex number. --- Complex space. --- Conjugacy class. --- Connected component (graph theory). --- Coprime integers. --- Cube root. --- Derivative. --- Diagonal matrix. --- Differential equation. --- Dimension (vector space). --- Discrete group. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Ellipse. --- Elliptic curve. --- Equation. --- Existential quantification. --- Fiber bundle. --- Finite group. --- First principle. --- Fundamental group. --- Gelfand. --- Holomorphic function. --- Hypergeometric function. --- Hyperplane. --- Hypersurface. --- Integer. --- Inverse function. --- Irreducible component. --- Irreducible representation. --- Isolated point. --- Isomorphism class. --- Line bundle. --- Linear combination. --- Linear differential equation. --- Local coordinates. --- Local system. --- Locally finite collection. --- Mathematical proof. --- Minkowski space. --- Moduli space. --- Monodromy. --- Morphism. --- Multiplicative group. --- Neighbourhood (mathematics). --- Open set. --- Orbifold. --- Permutation. --- Picard group. --- Point at infinity. --- Polynomial ring. --- Projective line. --- Projective plane. --- Projective space. --- Root of unity. --- Second derivative. --- Simple group. --- Smoothness. --- Subgroup. --- Subset. --- Symmetry group. --- Tangent space. --- Tangent. --- Theorem. --- Transversal (geometry). --- Uniqueness theorem. --- Variable (mathematics). --- Vector space.

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Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Ordered algebraic structures --- 512.74 --- Abelian varieties --- Modular functions --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties. --- Modular functions. --- Abelian extension. --- Abelian group. --- Abelian variety. --- Absolute value. --- Adele ring. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic equation. --- Algebraic extension. --- Algebraic number field. --- Algebraic structure. --- Algebraic variety. --- Analytic manifold. --- Automorphic function. --- Automorphism. --- Big O notation. --- CM-field. --- Characteristic polynomial. --- Class field theory. --- Coefficient. --- Complete variety. --- Complex conjugate. --- Complex multiplication. --- Complex number. --- Complex torus. --- Corollary. --- Degenerate bilinear form. --- Differential form. --- Direct product. --- Direct proof. --- Discrete valuation ring. --- Divisor. --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Existential quantification. --- Field of fractions. --- Finite field. --- Fractional ideal. --- Function (mathematics). --- Fundamental theorem. --- Galois extension. --- Galois group. --- Galois theory. --- Generic point. --- Ground field. --- Group theory. --- Groupoid. --- Hecke character. --- Homology (mathematics). --- Homomorphism. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Lie group. --- Linear combination. --- Linear subspace. --- Local ring. --- Modular form. --- Natural number. --- Number theory. --- Polynomial. --- Prime factor. --- Prime ideal. --- Projective space. --- Projective variety. --- Rational function. --- Rational mapping. --- Rational number. --- Real number. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scientific notation. --- Semisimple algebra. --- Simple algebra. --- Singular value. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Theorem. --- Vector space. --- Zero element.

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The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations.If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol.It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.

Operator theory --- Toeplitz operators --- Spectral theory (Mathematics) --- 517.984 --- Spectral theory of linear operators --- Toeplitz operators. --- Spectral theory (Mathematics). --- 517.984 Spectral theory of linear operators --- Operators, Toeplitz --- Linear operators --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Algebraic variety. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Change of variables. --- Chern class. --- Codimension. --- Cohomology. --- Compact group. --- Complex manifold. --- Complex vector bundle. --- Connection form. --- Contact geometry. --- Corollary. --- Cotangent bundle. --- Curvature form. --- Diffeomorphism. --- Differentiable manifold. --- Dimensional analysis. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Elaboration. --- Elliptic operator. --- Embedding. --- Equivalence class. --- Existential quantification. --- Exterior (topology). --- Fourier integral operator. --- Fourier transform. --- Hamiltonian vector field. --- Holomorphic function. --- Homogeneous function. --- Hypoelliptic operator. --- Integer. --- Integral curve. --- Integral transform. --- Invariant subspace. --- Lagrangian (field theory). --- Lagrangian. --- Limit point. --- Line bundle. --- Linear map. --- Mathematics. --- Metaplectic group. --- Natural number. --- Normal space. --- One-form. --- Open set. --- Operator (physics). --- Oscillatory integral. --- Parallel transport. --- Parameter. --- Parametrix. --- Periodic function. --- Polynomial. --- Projection (linear algebra). --- Projective variety. --- Pseudo-differential operator. --- Q.E.D. --- Quadratic form. --- Quantity. --- Quotient ring. --- Real number. --- Scientific notation. --- Self-adjoint. --- Smoothness. --- Spectral theorem. --- Spectral theory. --- Square root. --- Submanifold. --- Summation. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic manifold. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Todd class. --- Toeplitz algebra. --- Toeplitz matrix. --- Toeplitz operator. --- Trace formula. --- Transversal (geometry). --- Trigonometric functions. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Wave front set. --- Opérateurs pseudo-différentiels

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The description for this book, Linear Inequalities and Related Systems. (AM-38), Volume 38, will be forthcoming.

Operational research. Game theory --- Linear programming. --- Matrices. --- Game theory. --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Production scheduling --- Programming (Mathematics) --- Banach space. --- Basic solution (linear programming). --- Big O notation. --- Bilinear form. --- Boundary (topology). --- Brouwer fixed-point theorem. --- Characterization (mathematics). --- Coefficient. --- Combination. --- Computation. --- Computational problem. --- Convex combination. --- Convex cone. --- Convex hull. --- Convex set. --- Corollary. --- Correlation and dependence. --- Cramer's rule. --- Cyclic permutation. --- Dedekind cut. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Dilworth's theorem. --- Dimension (vector space). --- Directional derivative. --- Disjoint sets. --- Doubly stochastic matrix. --- Dual space. --- Duality (mathematics). --- Duality (optimization). --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation solving. --- Equation. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Extreme point. --- Fixed-point theorem. --- Functional analysis. --- Fundamental theorem. --- General equilibrium theory. --- Hall's theorem. --- Hilbert space. --- Incidence matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Invertible matrix. --- Kakutani fixed-point theorem. --- Lagrange multiplier. --- Linear equation. --- Linear inequality. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Loss function. --- Main diagonal. --- Mathematical induction. --- Mathematical optimization. --- Mathematical problem. --- Max-flow min-cut theorem. --- Maxima and minima. --- Maximal set. --- Maximum flow problem. --- Menger's theorem. --- Minor (linear algebra). --- Monotonic function. --- N-vector. --- Nonlinear programming. --- Nonnegative matrix. --- Parity (mathematics). --- Partially ordered set. --- Permutation matrix. --- Permutation. --- Polyhedron. --- Quantity. --- Representation theorem. --- Row and column vectors. --- Scientific notation. --- Sensitivity analysis. --- Set notation. --- Sign (mathematics). --- Simplex algorithm. --- Simultaneous equations. --- Solution set. --- Special case. --- Subset. --- Summation. --- System of linear equations. --- Theorem. --- Transpose. --- Unit sphere. --- Unit vector. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Von Neumann's theorem.