We consider complexifications of Riemannian symmetric spaces $X$ of
nonpositive curvature. We show that the maximal Grauert domain of $X$
is biholomorphic to a maximal connected extension $\Omega\sb {{\rm
AG}}$ of $X=G/K\subset G\sb {\mathbb {C}}/K\sb {\mathbb {C}}$ on which
$G$ acts properly, a domain first studied by D. Akhiezer and
S. Gindikin [1]. We determine when such domains are rigid, that is,
when ${\rm Aut}\sb {\mathbb {C}}(\Omega\sb {{\rm AG}}=G$ and when it
is not (when \Omega\sb {{\rm AG}}$ has "hidden symmetries"). We
further compute the $G$-invariant plurisubharmonic functions on
$\Omega\sb {{\rm AG}}$ and related domains in terms of Weyl group
invariant strictly convex functions on a $W$-invariant convex
neighborhood of $0\in \mathfrak {a}$. This generalizes previous
results of M. Lassalle [25] and others. Similar results have also been
proven recently by Gindikin and B. Krötz [8] and by
Krötz and R. Stanton [24].