On the embeddability of certain infinitely divisible probability measures on Lie groups
The embedding problem for infinitely divisible probabilities on locally compact groups was one of the most challenging problems during the last decades. The investigations started with pioneer papers, e.g., by \textit{E. Siebert} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 28, 227--247 (1974; Zbl 0264.60004)], followed by a series of investigations by S. G. Dani and M. McCrudden (see, e.g., [\textit{M. McCrudden}, in: S. G. Dani (ed.) et al., Probability measures on groups: recent directions and trends. Tata Institute of Fundamental Research 18, 331--363 (2006; Zbl 1145.60006); \textit{S. G. Dani}, Indian J. Pure Appl. Math. 41, No. 1, 241--250 (2010; Zbl 1200.60010)] and the references therein). It is a natural conjecture that the embedding problem has a positive solution for all connected Lie groups, but, for a long time, it was only possible to show this for connected Lie groups with nil-radical containing no torus of positive dimension. An example of a group not belonging to that class is the ``walnut group'' \(W\), a four dimensional Lie group, (cf., e.g., [\textit{S. G. Dani}, \textit{M. McCrudden} and \textit{S. Walker}, Math. Z. 245, No. 4, 781--790 (2003); erratum ibid. 252, No. 2, 457--458 (2006; Zbl 1050.22009)] for details). In the paper under review, the authors solve the embedding problem for a class of groups containing \(W\) (\(W\) was called walnut group because it seemed to be very hard to solve the embedding problem for that group) under an additional condition on the measure. Precisely, embeddability is shown for Lie groups \(G\) admitting a surjective continuous homomorphism \(p\) onto an almost algebraic group with \(\operatorname{ker} p \subseteq \operatorname{cent}(G)\), \(T:= \left(\operatorname{ker} p\right)^0\) being compact, and for infinitely divisible probabilities \(\mu\) whose image on \(G/T\) has no non-trivial idempotent factor (Theorem 1.3). The proof relies on standard techniques and methods used in the afore-mentioned investigations combined with recent results concerning central extensions of groups, measures on vector spaces \(V\) which are invariant under compact subgroups of \(\operatorname{GL}(V)\) and asymptotics of measures under certain automorphisms (called shear transformations). The tools developed there are interesting in their own right and will probably be useful in the future for further investigations.
scientific article; zbMATH DE number 6097111
scientific article