Factorization theorems on symmetric spaces of noncompact type
Let \(G\) be a semisimple noncompact Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). Denote by \(M^\#(X)\) the set of all probability measures on \(X= G/K\) which are invariant with respect to action of \(K\) on \(X\). The following analogues of well-known Khinchin's theorems are proved: Theorem 3.1. Every measure \(\mu\in M^\#(X)\) may be decomposed into \(\mu= (\nu_1*\nu_2*\cdots)* \omega\), where \(\nu_1,\nu_2,\dots\) is a finite or countable family of indecomposable \(K\)- invariant measures on \(X\) and \(\omega\) is a measure without indecomposable factors. Theorem 3.2. If \(\omega\) in \(M^\#(X)\) is a measure without indecomposable factors, then \(\omega\) is infinitely divisible. The main tool of the proofs of these theorems is the Kendall theory of delphic semigroups.
scientific article; zbMATH DE number 1309012