Semistable selfdecomposable laws on groups (Q2747065)

It is well known that for the vector spaces \(\mathbb{R}^d\), \(d\geq 1\), semistable laws are in general not selfdecomposable. \textit{A. Łuczak} [Probab. Theory Relat. Fields 90, No. 3, 317-340 (1991; Zbl 0734.60016)] described the intersection of these classes for vector spaces. The authors continue, in particular, these investigations. Their aim is to show that semistable selfdecomposable laws on a simple connected nilpotent Lie group correspond to (strictly) operator-semistable selfdecomposable (or operator Lévy's) measures on the tangent space and vice versa. It follows from this the complete analogue of Łuczak's characterization for simply connected nilpotent Lie groups.