Integral expression of some indecomposable characters of the infinite symmetric group in terms of irreducible representations (Q1121987)

Let \({\mathfrak S}_{\infty}\) be the infinite symmetric group and \({\mathfrak E}({\mathfrak S}_{\infty})\) the set of all indecomposable characters of \({\mathfrak S}_{\infty}\). E. Thoma established a one-to-one correspondence between \({\mathfrak E}({\mathfrak S}_{\infty})\) and the space \({\mathfrak Q}\) of sequences \(q=(q_ i)^{+\infty}_{i=-\infty}\) of non-negative numbers such that (i) \(q_ 1\geq q_ 2\geq...\); (ii) \(q_{-1}\geq q_{- 2}\geq...\); (iii) \(\sum^{+\infty}_{i=-\infty}q_ i=1\). In this paper the author shows that every indecomposable character \(\phi\in {\mathfrak E}({\mathfrak S}_{\infty})\) corresponding to \(q\in {\mathfrak Q}\) with \(q_ 0=0\) admits an integral expression in terms of positive definite functions which generate irreducible representations of \({\mathfrak S}_{\infty}\). These irreducible representations are obtained from ``infinite Young subgroups'' by means of inducing up their one- dimensional representations and the integral is taken over the space of ``infinite Young tableaux'' equipped with a probability measure.