Convolution roots and embeddings of probability measures on Lie groups

We show that for a large class of connected Lie groups G, viz. from classC described below, given a probability measure μ on G and a natural number n, for any sequence {νi} of th convolution roots of μ there exists a sequence {zi} of elements of G, centralising the support of μ, and such that is relatively compact; thus the set of roots is relatively compact ‘modulo’ the conjugation action of the centraliser of suppμ. We also analyse the dependence of the sequence {zi} on n. The results yield a simpler and more transparent proof of the embedding theorem for infinitely divisible probability measures on the Lie groups as above, proved in [S.G. Dani, M. McCrudden, Embeddability of infinitely divisible distributions on linear Lie groups, Invent. Math. 110 (1992) 237-261].