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We introduce a two-parameter function $\phi_{q_+,q_-}$ on the infinitehyperoctahedral group, which is a bivariate refinement of the reflection lengthkeeping track of the long and the short reflections separately. We show thatthis signed reflection function $\phi_{q_+,q_-}$ is positive definite if andonly if it is an extreme character of the infinite hyperoctahedral group and weclassify the corresponding set of parameters $q_+,q_-$. We construct thecorresponding representations through a natural action of the hyperoctahedralgroup $B(n)$ on the tensor product of $n$ copies of a vector space, which givesa two-parameter analog of the classical construction of Schur--Weyl.  We apply our classification to construct a cyclic Fock space of type Bgeneralizing the one-parameter construction in type A found previously byBo\.zejko and Guta. We also construct a new Gaussian operator acting on thecyclic Fock space of type B and we relate its moments with theAskey--Wimp--Kerov distribution by using the notion of cycles onpair-partitions, which we introduce here. Finally, we explain how to solve theanalogous problem for the Coxeter groups of type D by using our main result.

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