Embeddability of infinitely divisible distributions on linear Lie groups
A probability measure \(\mu\) on a locally compact group \(G\) is said to be infinitely divisible if it admits an \(n\)-th root (in the sense of convolution) of any order \(n\geq 1\), and it is called embeddable if there exists a continuous convolution semigroup \((\mu_ t)_{t>0}\) of probability measures on \(G\) such that \(\mu=\mu_ 1\). Clearly, on any locally compact group \(G\) every embeddable measure is infinitely divisible. The inverse statement gives rise to the question for which locally compact groups \(G\) the embedding property (EP) holds. There is a long standing conjecture that all connected Lie groups admit (EP). The authors achieve the remarkable result that every connected Lie group \(G\) having a representation \(\rho:G\to GL(d,\mathbb{R})\) for some \(d\geq 1\) such that \(\ker\rho\) is discrete admits (EP). The deep and elaborate proof of this embedding theorem is reduced to the case of almost algebraic groups which by definition are subgroups of finite index of real algebraic groups (Theorem 5.4). Then by considering coverings of such groups it is shown that any semisimple connected Lie group admits (EP) (Corollary 6.3). Finally, proceeding to coverings of linear Lie groups the authors establish (EP) for all simply connected Lie groups (Corollary of 7).
scientific article; zbMATH DE number 167394
scientific article