Indecomposable characters of the wreath product of a torus by the infinite block-diagonal symmetric group

In this paper we obtain a complete description of all indecomposable characters (central positive-definite functions) of “block-diagonal” inductive limits of semidirect products ${\mathbb{T}}^N⋊{\mathfrak{S}}_N$ of the compact $N$-dimensional torus ${\mathbb{T}}^N$ with the symmetric group ${\mathfrak{S}}_N$. The classification is proved by means of a variant of the ergodic method of Vershik and Kerov, namely by studying the weak limits of characters of finite-dimensional subgroups. As an application of our results, we give another proof of the classification of indecomposable characters of the infinite unitary group with block diagonal embeddings.