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

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).

### 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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### References:

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