Compactifications of Symmetric and Locally Symmetric Spaces by Armand Borel

By Armand Borel

Noncompact symmetric and in the neighborhood symmetric areas evidently seem in lots of mathematical theories, together with research (representation thought, nonabelian harmonic analysis), quantity thought (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In such a lot purposes it will be significant to shape a suitable compactification of the distance. The literature facing such compactifications is enormous. the most goal of this ebook is to introduce uniform structures of many of the recognized compactifications with emphasis on their geometric and topological structures.

The publication is split into 3 components. half I experiences compactifications of Riemannian symmetric areas and their mathematics quotients. half II is a research of compact soft manifolds. half III experiences the compactification of in the neighborhood symmetric spaces.

Familiarity with the speculation of semisimple Lie teams is thought, as is familiarity with algebraic teams outlined over the rational numbers in later components of the publication, even if many of the pertinent fabric is recalled as offered. another way, the publication is a self-contained reference aimed toward graduate scholars and examine mathematicians attracted to the functions of Lie thought and illustration idea to assorted fields of mathematics.

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In the following, linear algebraic groups are denoted by boldface capital roman letters, and the corresponding Lie groups by capital roman letters. For example, G denotes a reductive linear algebraic group defined over Q, and G denotes its real locus G(R); a rational parabolic subgroup is often denoted by P or Q, while a real parabolic subgroup is denoted by P or Q. In this book, parabolic subgroups are always assumed to be proper unless indicated otherwise. For a rational parabolic subgroup P, its real locus P = P(R) has two Langlands decompositions: the real and the rational decompositions, P = NP AP MP , P = N P A P MP , where A P is the real split component, and AP is the rational split component, AP ⊆ A P with equality if and only if the Q-rank of P is equal to the R-rank of P.

14). It also leads to a decomposition of G. In fact, G = P K , which is equivalent to that P acts transitively on X = G/K . It can be shown that K ∩ P = K ∩ M P , which implies that G = NP AP MP K ∼ = NP × AP × MP K . 20) For applications to understanding relations between boundary components of different parabolic subgroups, we need the relative Langlands decomposition for pairs of parabolic subgroups. 22) called the relative horospherical decomposition for the pair P ⊂ Q. Since K Q = K ∩ M Q is a maximal compact subgroup and hence M Q = P (K ∩ M Q ), the Langlands decomposition of P implies the relative Langlands decomposition of Q with respect to P: Q = NP AP MP K Q ∼ = NP × AP × MP K Q.

Fix the maximal compact subgroup K = SO(2). Let X = SL(2, R)/SO(2) be the associated symmetric space. Then X has two important models: the upper half-plane H and the Poincar´e disk D. 1), the disk model is convenient for understanding the compactification of X and its boundary. On the other hand, the model H is convenient for understanding the structure of neighborhoods of cusps and the compactification of quotients \H. An effective and group-theoretical way to explain this upper-half-plane model H is to use the Langlands decomposition of parabolic subgroups.

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