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

Abstract

We introduce a two-parameter function ${\varphi }_{q_{+},q_{-}}$ 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 show that this signed reflection function ${\varphi }_{q_{+},q_{-}}$ 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 one-parameter 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.

31 pages, 9 figures, comments are welcome; v2: 36 pages, 9 figures; new results in Section 4.2.2; Appendix, where we discuss the analogous situation in type D (which completes the picture of all infinite Coxeter groups of Weyl type)

References in corpus (2)

• The combinatorics of associated Hermite polynomials
• Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surface

Cited by in corpus (3)

• Matrix addition and the Dunkl transform at high temperature
• On the Thermodynamics of Particles Obeying Monotone Statistics
• The Double Fock Space of Type B