The geometry of Grauert tubes and complexification of symmetric spaces. (English) Zbl 1044.53039

Burns, D.; Halverscheid, S.; Hind, R.

Duke Math. J. 118, No. 3, 465-491 (2003).

If \(M\) is a real analytic Riemannian manifold, a Grauert domain is a neighborhood of \(M\) in the tangent bundle \(TM\) on which a complex structure can be defined in such a way that, for any geodesic \(\gamma \) of \(M\), the map \[ {\mathbb C} \rightarrow TM, \quad s+it \mapsto \bigl(\gamma (s),t\gamma'(s)\bigr), \] is holomorphic. If \(M=G/K\) is a Riemannian symmetric space of non compact type, the authors prove that the maximal Grauert domain is biholomorphic to the Akhiezer-Gindikin domain \(D\subset G^{\mathbb C}/K^{\mathbb C}\). This domain can be described as follows. Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be the Cartan decomposition of \({\mathfrak g}=\text{Lie}(G)\), and let \({\mathfrak a}\subset {\mathfrak p}\) be a Cartan subspace. Then \[ D=G\exp (i\omega)\cdot o, \] where \(o=eK\) is the base point, and \[ \omega =\{H\in {\mathfrak a}\mid \forall \alpha \in \Sigma,\;| \alpha (H)| <\tfrac{\pi}{2}\}. \] For a rank one symmetric space, the maximal Grauert domain \(D\) is the Grauert tube of maximal radius. If the symmetric space \(M\) is Hermitian, then \(D\simeq M\times \overline M\). If \(M\) is a real form of a Hermitian symmetric space \(N\), then \(N\) is the maximal Grauert domain for \(M\) if and only if \(\text{ rank}(N)=2\text{ rank}(M)\). One says that the maximal Grauert domain \(D\) is rigid if \(\text{Aut}(D)\simeq \text{Isom}(M)\).
It is proved that either \(D\) is rigid or \(D\) is a Hermitian symmetric space. Further the authors prove that the maximal Grauert domain \(D\) is a Stein manifold. This has been conjectured by Akhiezer and Gindikin in 1990. The main step in the proof is to establish a bijection between \(G\)-invariant strictly pluriharmonic functions on \(D\) and \(W\)-invariant strictly convex functions on \(\omega \) (\(W\) is the Weyl group of the root system \(\Sigma \)). This fact was already known in the case of a compact symmetric space (Azod-Loeb, Lassalle).

Reviewer: Jacques Faraut (Paris)

MSC:

53C35	Differential geometry of symmetric spaces
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32Q28	Stein manifolds

Keywords:

Grauert domain; symmetric space; Stein manifold

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