The Geometry of Grauert Tubes and Complexification of Symmetric Spaces

Abstract

We study the canonical complexifications of non-compact Riemannian symmetric spaces G/K by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed in another context by Akhiezer and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show there is an alternative for a G-invariant complexification: it is either "rigid" (its automorphism group is G), or it is a Hermmitian symmetric space. We also determine when invariant complexifications, especially the maximal one, are Hermitian symmetric. This is expressed simply in terms of the ranks of the symmetric spaces involved.

31 pages, LaTeX