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

Abstract

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.

44 pages (including supplementary material), comments are welcome