变晶冰的杨-巴克斯特方程
A Yang-Baxter equation for metaplectic ice
摘要：
我们将给出量子群在研究$GL(r,F)$的亚群$n$ -折叠覆盖上的球Whittaker函数中的新应用，其中$F$是非阿基米德局域场.早些时候，Brubaker、Bump、Friedberg、Chinta和Gunnells已经证明，这些Whittaker函数可以用统计力学系统的分拆函数来识别。他们假设这些惠特克函数的性质是由杨-巴克斯特方程决定的。我们证实了这一点，并确定相应的杨-巴克斯特方程与量子仿射李超代数$U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$，修改德林费尔德扭曲引入高斯和。(The变形参数$v$专用于剩余字段基数的倒数。对于亚群的主列表示，Whittaker模型不是唯一的.标准缠绕算子的散射矩阵是向量值的。对于一个简单的反射，它是由Kazhdan和Patterson计算的，他们将其应用于广义theta级数。我们将证明，一个简单的反射的Whittaker函数空间上的散射矩阵与量子群$U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(n))$的扭曲$R$ -矩阵相一致。这是上面提到的$U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$的扭曲$R$矩阵的一部分。
英文摘要：
We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, mentioned above.
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