Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

Abstract

We introduce a two-parameter function $\phi_{q_+, q_-}$ on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length keeping track of the long and the short reflections separately. We provide a complete characterization of the parameters $q_+, q_-$ when the signed reflection function $\phi_{q_+, q_-}$ is positive definite and we prove that this condition holds if and only if $\phi_{q_+, q_-}$ is an extreme character of the infinite hyperoctahedral group. We construct the corresponding representations as a natural action of the hyperoctahedral group $B(n)$ on the tensor product of $n$ copies of a vector space, which gives a two-parameter analog of the classical construction of Schur–Weyl. We apply our characterization to construct a cyclic Fock space of type B which generalizes the one-parameter construction in type A found previously by Bożejko and Guta. We also construct a new cyclic Gaussian operator of type B and we relate its moments with the Askey–Wilson–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here.