Halverscheid, Stefan; Iannuzzi, Andrea A family of adapted complexifications for \(SL_2(\mathbb R)\). (English) Zbl 1180.53053 Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 8, No. 1, 17-49 (2009). 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. Reviewer: Gabriela Paola Ovando (Freiburg) Cited in 1 Document 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 Keywords:adapted complexifications; naturally reductive metrics; non compact semisimple Lie groups PDFBibTeX XMLCite \textit{S. Halverscheid} and \textit{A. Iannuzzi}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 8, No. 1, 17--49 (2009; Zbl 1180.53053) Full Text: DOI