The geometry of Grauert tubes and complexification of symmetric spaces

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)AG of X= G/K (subset) GC/KC 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 AutC((omega)AG)=G and when it is not (when (omega)AG has "hidden symmetries"). We further compute the Ginvariant plurisubharmonic functions on (omega)AG and related domains in terms of Weyl group invariant strictly convex functions on a Winvariant convex neighborhood of 0 (element of) a. This generalizes previous results of M. Lassalle [25] and others. Similar results have also been proven recently by Gindikin and B. Krotz [8] and by Krotz and R. Stanton [24].