Fock space associated to Coxeter groups of type B

Abstract

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an (α, q)-Fock space, which satisfy the commutation relationwhere x, y are elements of a complex Hilbert space with a self-adjoint involution x →x and N is the number operator with respect to the grading on the (α, q)-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators b α,q (x) + b * α,q (x) with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution is associated to the orthogonal polynomials called the q-Meixner-Pollaczek polynomials, yielding the q-Hermite polynomials when α = 0 and free Meixner polynomials when q = 0.

Type B Gaussian statistics as noncommutative central limits

“…The type B Gaussian elements of [BEH15] 'live' in the algebra of bounded linear operators on a deformed Fock space. We briefly outline their construction and its context.…”

Radial Bargmann representation for the Fock space of type B

“…Let us point out some of the keys to calculate the distribution of G α,q (f ) in [BEH15]. It is shown that a linear map, Φ :…”

“…The (α, q)-inner product on the full Fock space F (H) is defined by The readers can consult [BEH15] for details.…”

“…It suggests that the theory of orthogonal polynomials plays intrinsic roles in all previous works mentioned above. In fact, the measure µ α,q,f given in [BEH15,Theorem 3.3] is derived essentially from the orthogonality measure ν α,q associated with the q-Meixner-Pollaczek orthogonal polynomials {P (α,q) n (x)} for α, q ∈ (−1, 1) given by the recurrence relation, P (α,q) 0 (x) = 1, P (α,q) 1 (x) = x, xP (α,q) n (x) = P (α,q) n+1 (x) + (1 + αq n−1 )[n] q P (α,q) n−1 (x), n ≥ 1 where [n] q = 1 + q + · · ·+ q n−1 is the q number. However, the Bargmann representation (measure on C) of ν α,q has not been obtained yet except the case of α = 0 for 0 ≤ q < 1 [LM95], for q = 1 [Barg61][AKK03], for q = 0 [Bi97], and t-deformed cases of these [AKW16][ KW14], and for q > 1 [Kr98].…”

“…In Section 4, commutation relations satisfied by one-mode (α, q)-annihilation and creation operators will be treated. In the Appendix, we shall give a minimum reference on the Coxeter group of type B extracted from [BEH15].…”