Satake compactifications.

*(English)*Zbl 0565.22009In 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.

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.

Reviewer: Gopal Prasad (Bombay)

##### MSC:

22E40 | Discrete subgroups of Lie groups |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

32J05 | Compactification of analytic spaces |