Exponential actions and maximal \({\mathfrak D}\)-invariant ideals (Q1387916)

It is shown that the closure of any orbit of an action of a connected, simple connected solvable exponential Lie group (associated with an exponential module with respect to the Lie algebra) contains at least a closed orbit. Furthermore a Wiener-like property is proved for nilpotent Lie groups with respect to an exponential action (see also \textit{H. Leptin} [Invent. Math. 31, 259-278 (1976; Zbl 0328.22012)], \textit{J. Ludwig} [Minimal \(C^*\)-dense ideals and algebraically irreducible representations of the Schwartz algebra of a nilpotent Lie group, in: Lect. Notes Math. 1359, 209-217 (1988; Zbl 0673.22001)], \textit{H. Leptin} and \textit{D. Poguntke} [J. Funct. Anal. 33, 119-134 (1979; Zbl 0414.43004)], \textit{J. Ludwig} [Math. Ann. 262, 287-304 (1983; Zbl 0501.22012)]).