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A family of adapted complexifications for \(SL_2(\mathbb R)\). (English) Zbl 1180.53053

Let \(G\) denote a connected non compact real semisimple Lie group equipped with a given naturally reductive left invariant pseudo Riemannian metric \(\nu_m\) associated to the decomposition \(\mathfrak g = \mathfrak k \oplus \mathfrak p\), being \(\mathfrak k\) a maximal compact subalgebra. The main goal of the paper is to present new examples of maximal complexifications adapted, in the non degenerate cases, to the Levi Civita connection for \(\nu_m\). In the degenerate cases one finds a maximal complexification adapted to the unique real analytic linear connection which is obtained as the limit of such Levi Civita connections.
The authors show the existence of a maximal complexification \(\Omega_m\) adapted to \(\nu_m\) which is realized as an \(L\)-equivariant Riemannian domain over the universal complexification \(G^{\mathbb C}\) of \(G\), with a polar map \(P_m:\Omega_m\to G^{\mathbb C}\). By considering the usual identification \(TG \simeq G \times \mathfrak g\) such complexification can be described via a slice for the induced \(L\)-action by \(\Omega_m=L \cdot \Sigma_m\), where \(\Sigma_m \subset \{e\}\times \mathfrak g\) is a semi-analytic subset of the product of \(\mathfrak k\) and the closure of the Weyl chamber in a maximal abelian subalgebra \(\mathfrak a\) of \(\mathfrak p\). The case of \(G=SL_2(\mathbb R)\) is worked out in detail.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C22 Geodesics in global differential geometry
32C09 Embedding of real-analytic manifolds
32Q99 Complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
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