A description of characters on the infinite wreath product

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $\Gamma^\infty$ by automorphisms, so one can form a semidirect product $\Gamma^\infty\rtimes \mathfrak{S}_\infty$, known as the {\it wreath} product $\Gamma\wr\mathfrak{S}_\infty$ of $\Gamma$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $II_1-$factor-representations of $\Gamma\wr\mathfrak{S}_\infty$ in terms of finite characters of $\Gamma$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty\times\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $II_1$, where the {\it modular operator} of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.