Fock space associated to Coxeter groups of type B

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書誌事項

公開日
2015-09
権利情報
  • https://www.elsevier.com/tdm/userlicense/1.0/
  • http://www.elsevier.com/open-access/userlicense/1.0/
DOI
  • 10.1016/j.jfa.2015.06.026
  • 10.48550/arxiv.1411.7997
公開者
Elsevier BV

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説明

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 $(��,q)$-Fock space, which satisfy the commutation relation $$ b_{��,q}(x)b_{��,q}^\ast(y)-qb_{��,q}^\ast(y)b_{��,q}(x)=\langle x, y\rangle I+��\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(��,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{��,q}(x)+b_{��,q}^\ast(x)$ 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 $��=0$ and free Meixner polynomials when $q=0$.

22 pages, 6 figures

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