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.

中文翻译：

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变形冰的 Yang-Baxter 方程

我们将给出量子群的新应用，用于研究 $\GL(r,F)$ 的元折元 $n$-fold 覆盖上的球形 Whittaker 函数，其中 $F$ 是非阿基米德局部场。早先的 Brubaker、Bump、Friedberg、Chinta 和 Gunnells 已经证明，这些 Whittaker 函数可以与统计力学系统的分区函数相识别。他们假设 Yang-Baxter 方程是这些 Whittaker 函数性质的基础。我们证实了这一点，并确定了对应的 Yang-Baxter 方程与量子仿射李超代数 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$，由 Drinfeld 修改扭曲以引入高斯和。（变形参数 $v$ 专门用于残差场基数的倒数。）对于 metaplectic 群的主级数表示，Whittaker 模型并不是独一无二的。标准交织算子的散射矩阵是向量值。对于简单的反射，它由 Kazhdan 和 Patterson 计算，并将其应用于广义 theta 级数。我们将证明 Whittaker 函数空间上简单反射的散射矩阵与量子群 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}( n))$。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。我们将证明 Whittaker 函数空间上简单反射的散射矩阵与量子群 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}( n))$。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。我们将证明 Whittaker 函数空间上简单反射的散射矩阵与量子群 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}( n))$。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。