Lie semigroups with triple decompositions
The paper under review deals with order structures on symmetric spaces of the form \(G/H\) and the corresponding subsemigroups of \(G\). To describe the first main result, let \((G, \tau)\) be an involutive Lie group, i.e., \(\tau\) is an involutive automorphism of \(G\), and \(\mathbf g = \mathbf h \oplus \mathbf q\) the corresponding eigenspace decomposition of \(\mathbf g\) with respect to the differential of \(\tau\). Further let \(C\subseteq {\mathbf q}\) be a pointed generating closed convex cone which is invariant under the closed subgroup \(H\) with Lie algebra \(\mathbf h\). Then the \(G\)-invariant order structure on the homogeneous space \(G/H\) for which \(\exp(C) H\) is the set of elements greater than the base point \(e H\) is globally hyperbolic in the sense that all order intervals \([x,y] := \{ z \in G/H \: x \leq z \leq y\}\) are compact. This result was known before for more special situations [see the monograph of \textit{J. Hilgert} and \textit{G. Ólafsson}, ``Causal symmetric spaces'' (San Diego 1997; Zbl 0931.53004)] and the proof of the present general version is based on the machinery developed in [\textit{B. Krötz} and \textit{K.-H. Neeb}, J. Lie Theory 6, 69-146 (1996; Zbl 0860.22004)]. For the second main result, suppose that \(\mathbf q = \mathbf q_+ \oplus \mathbf q_-\) holds for two \(H\)-invariant abelian subalgebras \({\mathbf q}_\pm\) and that \(C_{\pm} \subseteq {\mathbf q}_\pm\) are pointed generating \(H\)-invariant closed convex cones. Then the authors give mild sufficient conditions for the multiplication map \[ C_- \times H \times C_+ \to G, \quad (X,h,Y) \mapsto \exp X \cdot h \cdot \exp Y \] to be a diffeomorphism onto a closed subsemigroup of \(G\). The latter result applies in particular to the compression semigroup of a symmetric cone.
scientific article; zbMATH DE number 1537839