遵循[Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378]和[Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588]的研究，我们对A_{n-1}型Kleinian奇点的量子化上的扭曲迹进行了详细研究。特别地，我们给出了这些迹的显式积分公式，并利用它们确定了一个迹何时在相应的代数上定义一个正定的埃尔米特形式。这导致了此类量子化的幺正短星积的分类，这是Beem、Peelaers和Rastelli在与三维超共形场论相关联时提出的一个问题。特别地，我们证实了他们的猜想：对于n≤4，幺正短星积是唯一的，并将其参数作为量子化参数的函数进行了计算，给出了Beem、Peelaers和Rastelli所提数值函数的精确公式。若n=2，这特别地恢复了sl_2的幺正球面Harish-Chandra双模理论。因此，本文的结果可被视为将约化李代数包络代数上的幺正Harish-Chandra双模理论[Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987]推广到更一般量子代数的起点。最后，我们推导了用于计算与扭曲迹对应的短星积系数的递推关系，这些递推关系是离散Painlevé系统的推广。
英文摘要：
Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.
Quantized Howe-type dualities via Koornwinder polynomials and the X=K phenomenon
Institut Camille Jordan, Lyon 1 University（里昂第一大学）；Department of Mathematics, University of Pittsburgh（匹兹堡大学）；Institut Denis Poisson, University of Tours（图尔大学）；Department of Mathematics and Statistics, State University of New York at Albany（纽约州立大学奥尔巴尼分校）
