Items related to Compactifications of Symmetric and Locally Symmetric...
Synopsis
Non-compact Riemannian symmetric spaces and their quotients by arithmetically defined groups of isometries occur in many parts of mathematics. For many purposes it has been necessary to compactify them. This monograph attempts to give a systematic, and in part new exposition of these compactifications, of new ones, their interrelations and of the context out of which they arose. Compactifications of the more general semisimple symmetric spaces are also considered. The book is divided into three main parts. Part I is devoted to five types of compactifications, some related even isomorphic, all G-spaces, of the quotient X=3DG/K of a semisimple linear real Lie group G with finitely many connected components by a maximal compact subgroup K. The second part treats compactifications of the quotients GammaX, where Gamma is an arithmetic subgroup of G, assumed to be defined over the field Q of the rational numbers. In the third part, three new types of compactifications are the direct constructions of T. Oshima and T. Oshima--T. Sekiguchi, the gluing of a certain number of copies of a compact manifold with corners, and the real points of the so-called wonderful compactification of the complexification of X, or more generally G/H. The compactification of noncompact Riemannian symmetric spaces leads to a rich area of research in which many mathematical disciplines come algebraic topology, geometry, number theory and representation theory. Familiarity with the theory of real semisimple Lie groups and symmetric spaces, and in Part II, of linear algebraic groups over Q is assumed although much material is recalled along the way. Of interest and use to researchersand graduate students in Lie Theory or Representation Theory.
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