On representations of the infinite symmetric group

We prove a classification theorem for admissible representation of the Gelfand pair
$$ S(\infty)\times S(\infty)\supset\operatorname{diag}S(\infty) $$
and two other Gelfand pairs of hyperoctohedral type. We prove that the list of admissible representations given by G. Olshanski is complete. This generalizes Thoma's description of the characters of $S(\infty)$. An explicit construction for representations from a dense subset of the admissible dual was given by G. Olshanski. We construct the remaining representations using an operation we call the mixture of representations.