Satake compactifications
type: Item
instance of: scholarly article
title: Satake compactifications
zbMATH Open document ID: 0565.22009
DOI: 10.1007/BF02564638
author: Steven Zucker
published in: Commentarii Mathematici Helvetici
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139942
review text: In a fundamental paper [Comment. Math. Helv. 48, 436--491 (1973; Zbl 0274.22011)] \textit{A. Borel} and \textit{J. P. Serre} constructed a compactification of the locally symmetric space \(\Gamma\setminus X\); where \(\Gamma\) is an arithmetic subgroup of the group \(G\) of isometries of a symmetric space \(X\) with nonpositive sectional curvatures. The main result of the paper under review is that the Satake compactification of \(\Gamma\setminus X\) for certain representations \(\bullet\) (for example if \(\tau\) is defined over \(Q\)) is a quotient of the Borel-Serre compactification. Inspired by the construction of Borel and Serre of the corner \(X(P)\) associated with a parabolic \(Q\)-subgroup \(P\), given a finite dimensional representation \(\bullet\) of \(G\), the author constructs the ''crumpled corner'' \(X^*(P)\) which is a quotient of \(X(P)\). For parabolic \(Q\)-subgroups \(P\subset Q\), \(X^*(Q)\) is embedded in \(X^*(P)\) as an open subset and this embedding is compatible with the projections \(X(P)\to X^*(P)\) and \(X(Q)\to X^*(Q)\). Let \({}_Q\tilde X^*\) be the union of the \(X^*(P)'s\). Then \({}_Q\tilde X^*\) is a quotient of the manifold \(\bar X\) with corners. The construction of the crumpled corner is such that it is seen at once that there is a natural bijection of \({}_Q\tilde X^*\) onto \({}_ QX^*_{\tau}\). It is proved here that this bijection is continuous. Now it follows immediately that the Satake compactification \(\Gamma \setminus_ QX^*_{\tau}\) is a quotient of the Borel-Serre compactification \(\Gamma\setminus \bar X\). If \(X\) is Hermitian \textit{W. L. Baily} jun. and \textit{A. Borel} [Ann. Math. (2) 84, 442--528 (1966; Zbl 0154.08602)] gave a compactification of \(\Gamma\setminus X\) which is a normal analytic space. It is shown in this paper that the Baily-Borel compactification is homeomorphic to the Satake compactification with respect to any representation of \(G\) whose restricted highest weight is a multiple of the dist…
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 32J05
zbMATH DE Number: 3901063
zbMATH Keywords: compactification
zbMATH Keywords: locally symmetric space
zbMATH Keywords: Satake compactification
zbMATH Keywords: Borel-Serre compactification
zbMATH Keywords: crumpled corner
zbMATH Keywords: Baily-Borel compactification
MaRDI profile type: Publication
OpenAlex ID: W3188616406
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description: scientific article; zbMATH DE number 3901063