具有两个参数的反射长度在类型B/C的渐近表示论中的应用
Reflection length with two parameters in the asymptotic representation theory of type B/C and applications
AI总结：
本文引入了一个双参数函数，用于跟踪长和短反射，应用于构造B型Fock空间和新的高斯算子，同时关联了Askey-Wimp-Kerov分布。
摘要：
我们引入了一个双参数函数$ϕ_{q_+,q_-}$，该函数是对无限超立方群的反射长度的双变量细化，分别跟踪长和短反射。我们证明该符号反射函数$ϕ_{q_+,q_-}$正定当且仅当它是无限超立方群的极值特征，并分类了相应的参数集$q_+,q_-$.我们通过超立方群$B(n)$在向量空间张量积上的自然作用构造了相应的表示，这给出了Schur-Weyl经典构造的双参数类比。我们应用分类来构造B型的循环Fock空间，推广了之前Bożejko和Guta在A型中的一参数构造。我们还构造了一个新的高斯算子，作用于B型的循环Fock空间，并通过引入的配对划分上的循环概念，将其矩与Askey-Wimp-Kerov分布相关联。最后，我们解释了如何利用主要结果解决D型Coxeter群的类似问题。
英文摘要：
We introduce a two-parameter function $ϕ_{q_+,q_-}$ 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 $ϕ_{q_+,q_-}$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
