On Realizations of Representations of the Infinite Symmetric Group

Denote by $\mathbb{N}$ the set of positive integers {1, 2,…}. Let ${{\mathfrak{S}}_{\mathbb{X}}}$ stand for the group of all finite permutations of the set $\mathbb{X}=-\mathbb{N}\cup \mathbb{N}$. Consider the subgroups ${{\mathfrak{S}}_{\mathbb{N}}}=\left\{ {s\in {{\mathfrak{S}}_{\mathbb{X}}}:\;s\left( {-k} \right)=-k\;{\rm{for}}\;{\rm{all}}\;k\in \mathbb{N}} \right\}$ and $\mathfrak{D}=\left\{ {s\in {{\mathfrak{S}}_{\mathbb{X}}}: - s(k)=s\left( {-k} \right)\;\;{\rm{and}}\;\;s\left( \mathbb{N} \right)=\mathbb{N}} \right\}$. Given a spherical representation π of the pair $\left( {{{\mathfrak{S}}_{\mathbb{N}}}\cdot {{\mathfrak{S}}_{{-\mathbb{N}}}},\mathfrak{D}} \right)$, we construct a spherical representation Π of the pair $\left( {{{\mathfrak{S}}_{\mathbb{X}}},\mathfrak{D}} \right)$ such that the restriction of Π to the group ${{\mathfrak{S}}_{\mathbb{N}}}\cdot {{\mathfrak{S}}_{{-\mathbb{N}}}}$ coincides with π. Bibliography: 6 titles.