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

We introduce a two-parameter function φq+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function φq+,q− is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters q+, q−. We construct the corresponding representations through 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 classification to construct a cyclic Fock space of type B generalizing the oneparameter construction in type A found previously by Bożejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey–Wimp–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.

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