Compactifications of Symmetric and Locally Symmetric Spaces (Mathematics: Theory & Applications)

Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. Otherwise, the book is a self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics. Cover......Page 1 Compactifications of Symmetric and Locally Symmetric Spaces......Page 4 Copyright - ISBN: 0817632476......Page 5 Contents......Page 8 Preface......Page 12 0 Introduction......Page 18 0.1 History of compactifications......Page 19 0.2 New points of view in this book......Page 31 0.3 Organization and outline of the book......Page 33 0.4 Topics related to the book but not covered and classification of references......Page 34 Part I: Compactifications of Riemannian Symmetric Spaces......Page 40 1 Review of Classical Compactifications of Symmetric Spaces......Page 44 I.1 Real parabolic subgroups......Page 45 I.2 Geodesic compactification and Tits building......Page 56 I.3 Karpelevič compactification......Page 67 I.4 Satake compactifications......Page 72 I.5 Baily-Borel compactification......Page 94 I.6 Furstenberg compactifications......Page 108 I.7 Martin compactifications......Page 114 2 Uniform Construction of Compactifications of Symmetric Spaces......Page 124 I.8 Formulation of the uniform construction......Page 125 I.9 Siegel sets and generalizations......Page 131 I.10 Uniform construction of the maximal Satake compactification......Page 140 I.11 Uniform construction of nonmaximal Satake compactifications......Page 146 I.12 Uniform construction of the geodesic compactification......Page 156 I.13 Uniform construction of the Martin compactification......Page 161 I.14 Uniform construction of the Karpelevič compactification......Page 166 I.15 Real Borel-Serre partial compactification......Page 176 I.16 Relations between the compactifications......Page 182 I.17 More constructions of the maximal Satake compactification......Page 185 I.18 Compactifications as a topological ball......Page 191 I.19 Dual-cell compactification and maximal Satake compactification as a manifold with corners......Page 199 Part II: Smooth Compactifications of Semisimple Symmetric Spaces......Page 216 4 Smooth Compactifications of Riemannian Symmetric Spaces G/K......Page 220 II.1 Gluing of manifolds with corners......Page 221 II.2 The Oshima compactification of G/K......Page 227 II.3 Generalities on semisimple symmetric spaces......Page 232 II.4 Some real forms H[sub(ε)] of H......Page 234 II.5 Galois Cohomology......Page 238 II.6 Orbits of G in (G/H)(R)......Page 241 II.7 Examples......Page 247 II.8 Generalities on semisimple symmetric spaces......Page 250 II.9 The DeConcini-Procesi wonderful compactification of G/H......Page 253 II.10 Real points of G/H......Page 256 II.11 A characterization of the involutions σ[sub(ε)]......Page 261 II.12 Preliminaries on semisimple symmetric spaces......Page 266 II.13 The Oshima-Sekiguchi compactification of G/K......Page 270 II.14 Comparison with \overline{G/H}^W\mathbb{R}......Page 274 Part III: Compactifications of Locally Symmetric Spaces......Page 280 9 Classical Compactifications of Locally Symmetric Spaces......Page 284 III.1 Rational parabolic subgroups......Page 285 III.2 Arithmetic subgroups and reduction theories......Page 294 III.3 Satake compactifications of locally symmetric spaces......Page 303 III.4 Baily-Borel compactification......Page 310 III.5 Borel-Serre compactification......Page 318 III.6 Reductive Borel-Serre compactification......Page 326 III.7 Toroidal compactifications......Page 331 10 Uniform Construction of Compactifications of Locally Symmetric Spaces......Page 340 III.8 Formulation of the uniform construction......Page 341 III.9 Uniform construction of the Borel-Serre compactification......Page 343 III.10 Uniform construction of the reductive Borel-Serre compactification......Page 355 III.11 Uniform construction of the maximal Satake compactification......Page 362 III.12 Tits compactification......Page 367 III.13 Borel-Serre compactification of homogeneous spaces Γ\G......Page 372 III.14 Reductive Borel-Serre compactification of homogeneous spaces Γ\G......Page 376 11 Properties of Compactifications of Locally Symmetric Spaces......Page 382 III.15 Relations between the compactifications......Page 383 III.16 Self-gluing of Borel-Serre compactification into Borel-Serre-Oshima compactification......Page 387 12 Subgroup Compactifications of Γ\G......Page 392 III.17 Maximal discrete subgroups and space of subgroups......Page 393 III.18 Subgroup compactification of Γ\G and Γ\X......Page 399 III.19 Spaces of flags in \mathbb{R}[sup(n)], flag lattices, and compactifications of SL(n, \mathbb{Z})\SL(n,\mathbb{R})......Page 403 13 Metric Properties of Compactifications of Locally Symmetric Spaces Γ\X......Page 416 III.20 Eventually distance-minimizing geodesics and geodesic compactification of Γ\X......Page 417 III.21 Rough geometry of Γ\X and Siegel conjecture on metrics on Siegel sets......Page 422 III.22 Hyperbolic compactifications and extension of holomorphic maps from the punctured disk to Hermitian locally symmetric spaces......Page 426 III.23 Continuous spectrum, boundaries of compactifications, and scattering geodesics of Γ\X......Page 433 References......Page 440 N......Page 468 X......Page 469 B......Page 476 C......Page 478 D......Page 481 F......Page 482 G......Page 483 H......Page 484 L......Page 485 M......Page 486 P......Page 488 R......Page 490 S......Page 492 T......Page 495 Z......Page 496

Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures.

The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces.

Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. Otherwise, the book is a self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics.

"The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. Otherwise, the book is a self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics."--Jacket Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology