Meromorphic reduction of complex Lie groups (Q2772932)

\textit{A. Morimoto} proved [Proc. Conf. Complex Analysis, Minneapolis 1964, 256-272 (1965; Zbl 0144.07902)] that every connected complex Lie group \(G\) contains a closed, connected, complex, normal subgroup \(G_{0}\) called its Steinizer such that \(G/G_{0}\) is Stein and \(G_{0}\) has no non-constant holomorphic functions. Such a group \(G_{0}\) is called a toroidal group or Cousin group and has the form \({\mathbb C}^n/\Lambda\) for some discrete subgroup \(\Lambda \subset{\mathbb C}^n\). NEWLINENEWLINENEWLINEThe meromorphic reduction of toroidal groups was investigated by the present author [J. Math. Soc. Japan 41, 699-708 (1989; Zbl 0684.32020) and Ann. Mat. Pura Appl., IV. Ser. 169, 1-33 (1995; Zbl 0847.14030)], where it was shown that there exists a closed complex subgroup \(M \subset T = { \mathbb C}^n/\Lambda\) such that \(T/M\) is meromorphically separable and all meromorphic functions on \(T\) are pull-backs from \(T/M\) via the natural map \(T\to T/M\). Given a group \(G\) with Steinizer \(G_{0}\) let \(M\) be the corresponding group such that \(G_{0} \to G_{0}/M\) is the meromorphic reduction of the toroidal group \(G_{0}\). The author proves that \(G \to G/M\) is the meromorphic reduction of \(G\).