The geometry of Grauert tubes and complexification of symmetric spaces

摘要

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 of G(C)/K-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 Aut(C)(Omega(AG)) = G and when it is not (when Omega(AG) has "hidden symmetries "). We further compute the G-invariant plurisubharmonic functions on Omega(AG) and related domains in terms of Weyl group invariant strictly convex functions on a W invariant convex neighborhood of 0 is an 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]. [References: 38]