  Satake has constructed compactifications of symmetric spaces D=G/K which
(under a condition called geometric rationality by Casselman) yield
compactifications of the corresponding locally symmetric spaces. The different
compactifications depend on the choice of a representation of G. One example is
the Baily-Borel-Satake compactification of a Hermitian locally symmetric space;
Baily and Borel proved this is always geometrically rational. Satake
compactifications for which all the real boundary components are equal-rank
symmetric spaces are a natural generalization of the Baily-Borel-Satake
compactification. Recent work (see math.RT/0112250, math.RT/0112251) indicates
that this is the natural setting for many results about cohomology of
compactifications of locally symmetric spaces. In this paper we prove any
Satake compactification for which all the real boundary components are
equal-rank symmetric spaces is geometrically rational aside from certain
rational rank 1 or 2 exceptions; we completely analyze geometric rationality
for these exceptional cases. The proof uses Casselman's criterion for geometric
rationality. We also prove that a Satake compactification is geometrically
rational if the representation is defined over the rational numbers.
