The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

[KLM1] M. Kapovich, B. Leeb, J. J. Millson, Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity, Preprint, June 2004. [KLM2) M. Kapovich, B. Leeb, ...

Author: Michael Kapovich

Publisher: American Mathematical Soc.

ISBN: 9780821840542

Category: Mathematics

Page: 83

View: 493

In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the ""Saturation Conjecture"" for $GL(\ell)$ as a consequence of their solution of the corresponding ""saturation problem"" for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.
Categories: Mathematics

Classification of Generalized Affine Symmetric Spaces of Dimension N less Than Or Equal to Symbol 4

Classification of Generalized Affine Symmetric Spaces of Dimension N  less Than Or Equal to Symbol  4

[ 11 ] Generalized iffine symmetric spaces , Math . Nachr . 80 ( 1977 ) , pp . 205–208 . [ 12 ] Classification of generalized symmetric Riemannian spaces of dimension n < 5 , Rozpravy CSAV 85 ( 8 ) , ( 1975 ) .

Author: S. Węgrzynowski

Publisher:

ISBN: STANFORD:36105031977965

Category: Generalized spaces

Page: 80

View: 731

Categories: Generalized spaces

Eigenfunction Expansions Operator Algebras and Riemannian Symmetric Spaces

Eigenfunction Expansions  Operator Algebras and Riemannian Symmetric Spaces

... component of the isometry group on a symmetric space in terms of generalized eigenprojections with the same property , and the theory of the monograph indicates that the starting point must be a commutative von Neumann algebra .

Author: Robert M Kauffman

Publisher: CRC Press

ISBN: 0582276349

Category: Mathematics

Page: 152

View: 747

This Research Note pays particular attention to studying the convergence of the expansion and to the case where D is a family of partial differential operators. All operators in the natural von Neumann algebraassociated with D, and also unbounded operators affiliated with this algebra, are expanded simultaneously in terms of generalized eigenprojections. These are operators which carry a natural space associated with D into its dual. The elements of the range of these eigenprojections are the eigenfunctions, which solve the appropriate eigenvalue equations by duality. The spectral measure is abstractly defined, but its absolute continuity with respect to Hausdorf measure on the joint spectrum is shown to occur when the eigenfunctions are very well-behaved. Uniqueness results are given showing that any two expansions arise from each other by a simple change of variable. A considerable effort has been made to keep the book self-contained for readers with a background in functional analysis including a basic understanding of the theory of von Neumann algebras. More advanced topics in functional analysis, andan introduction to differential geometry and differential operator theory, mostly without proofs, are given in an extensive section on background material.
Categories: Mathematics

Compactifications of Symmetric Spaces

Compactifications of Symmetric Spaces

... ni e N', and a = a2a1 with a2 e A1, al e A'. Since AI commutes with N', na: o = n.2nia: o = n2a2n1a1 o = n2a2 a ', where x' is a point in the symmetric space X". The generalized horocyclic coordinates of x are then n2 ...

Author: Yves Guivarc'h

Publisher: Springer Science & Business Media

ISBN: 9781461224525

Category: Mathematics

Page: 286

View: 765

The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.
Categories: Mathematics

Differential Geometry Lie Groups and Symmetric Spaces over General Base Fields and Rings

Differential Geometry  Lie Groups and Symmetric Spaces over General Base Fields and Rings

900 899 898 897 896 895 894 893 892 891 890 889 888 887 886 885 884 883 882 881 880 879 878 877 876 Wolfgang Bertram, Differential geometry, Lie groups and symmetric spaces over general base fields and rings, 2008 Piotr Hajlasz, ...

Author: Wolfgang Bertram

Publisher: American Mathematical Soc.

ISBN: 9780821840917

Category: Mathematics

Page: 202

View: 114

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.
Categories: Mathematics

Harmonic Analysis on Symmetric Spaces Higher Rank Spaces Positive Definite Matrix Space and Generalizations

Harmonic Analysis on Symmetric Spaces   Higher Rank Spaces  Positive Definite Matrix Space and Generalizations

Siegel upper half space is identifiable with a bounded symmetric domain, namely the generalized unit disc: Dn D ̊ W2 Cnn ˇ ˇ tW D W; I WW 2 Pn « : (2.15) The identification map is the generalized Cayley transform: ̨ W Hn ! Dn Z! .Z iI/.

Author: Audrey Terras

Publisher: Springer

ISBN: 9781493934089

Category: Mathematics

Page: 487

View: 279

This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups Γ (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant ±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.
Categories: Mathematics

Symmetric Spaces and the Kashiwara Vergne Method

Symmetric Spaces and the Kashiwara Vergne Method

With the above tools at hand we can now try to extend to general symmetric spaces Duflo's isomorphism for Lie groups (Sect. 1.1). By Theorem 3.9 the transfer map e Á is an isomorphism of S .s/H; onto .D.S/;ı/, and the problem is to ...

Author: François Rouvière

Publisher: Springer

ISBN: 9783319097732

Category: Mathematics

Page: 196

View: 964

Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit of Kashiwara and Vergne's original work for Lie groups. The book includes a complete rewriting of several articles by the author, updated and improved following Alekseev, Meinrenken and Torossian's recent proofs of the conjecture. The chapters are largely independent of each other. Some open problems are suggested to encourage future research. It is aimed at graduate students and researchers with a basic knowledge of Lie theory.
Categories: Mathematics