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Satake compactifications. (English) Zbl 0565.22009
In a fundamental paper [Comment. Math. Helv. 48, 436–491 (1973; Zbl 0274.22011)] A. Borel and 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 W. L. Baily jun. and 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 distinguished fundamental dominant weight. As a consequence, it follows that the Baily-Borel compactification is also a quotient of the Borel-Serre compactification.

##### MSC:
 22E40 Discrete subgroups of Lie groups 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 32J05 Compactification of analytic spaces
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