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The Geometry of Grauert Tubes and Complexification of Symmetric Spaces

D. Burns·S. Halverscheid·R. Hind
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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.

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D. Burns, S. Halverscheid, R. Hind. 2001-09-24. The Geometry of Grauert Tubes and Complexification of Symmetric Spaces. https://arxiv.org/abs/math/0109186

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