Minimal and maximal semigroups related to causal symmetric spaces
Let \(G/H\) be an irreducible globally hyperbolic semisimple symmetric space. In the paper under review it is shown that each subsemigroup \(S \subseteq G\) with non-empty interior containing \(H\) in a non-isolated way, lies between two semigroups \(S_{\text{ min}} = H \exp(C_{\text{ min}})\) and \(S_{\text{ max}} = H Z_K({\mathbf a}) \exp(C_{\text{ max}})\), where \(C_{\text{ min}}\), resp., \(C_{\text{ max}}\) are minimal, resp., maximal non-trivial \(H\)-invariant cones in the tangent space of the origin of \(G/H\). From that one immediately obtains a description of the maximal semigroups with this property as \(S_{\text{ max}}\) and \(S_{\text{ max}}^{-1}\). The latter result was first obtained in [\textit{J. Hilgert} and \textit{K.-H. Neeb}, Semigroup Forum 50, 205-222 (1995; Zbl 0824.22006)] and later simplified for the case \(G/H \cong H_C/H\) by \textit{J.~D.~Lawson} [J. Lie Theory 4, 17-29 (1994; Zbl 0824.22012)]. In the present paper the authors follow essentially Lawson's strategy, and for that they need a good description of the \(H\)-double cosets in \(G\), namely that if \({\mathbf a}_1,\dots, {\mathbf a}_n\) are representatives of \(H\)-conjugacy classes of Cartan subspaces for \(G/H\), then \(\bigcup_{j = 1}^n H N_G({\mathbf a}_j) H\) contains an open dense subset of \(G\). This is the second main result of the paper. This result has recently been generalized to all semisimple symmetric spaces by \textit{A.~G.~Helminck, J.~Hilgert, A.~Neumann} and \textit{G.~Olafsson} [Math. Ann. 313, 785-791 (1999; Zbl 0926.22005)].
scientific article; zbMATH DE number 1491101