open access

https://hdl.handle.net/2115/76025

Fock space associated to Coxeter groups of type B

Fock spaces

2015-09-15

Journal of Functional Analysis

269(6)

1769-1795

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 relation $b_{α,q}(x)b^∗_{α,q}(y)−qb^∗_{α,q}(y)b_{α,q}(x)=⟨x,y⟩I+α⟨\overline{x},y⟩q^{2N}$, where x,y are elements of a complex Hilbert space with a self-adjoint involution $x↦\overline{x}$ 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 associates the orthogonal polynomials called the q-Meixner-Pollaczek polynomials, yielding the q-Hermite polynomials when α=0 and free Meixner polynomials when q=0.

学術雑誌論文

Accepted Manuscript

Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

0022-1236