A Yang–Baxter equation for metaplectic ice
A Yang–Baxter equation for metaplectic ice
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DOI：
10.4310/cntp.2019.v13.n1.a4
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发表时间：
2016-04
影响因子：
1.9
通讯作者：
Ben Brubaker;Valentin Buciumas;D. Bump
中科院分区：
文献类型：
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作者：
Ben Brubaker;Valentin Buciumas;D. Bump
我们将给出量子群的新应用，以研究 $\GL(r,F)$ 的元折 $n$ 倍覆盖上的球形 Whittaker 函数，其中 $F$ 是非阿基米德局部场。早些时候，Brubaker、Bump、Friedberg、Chinta 和 Gunnells 已经证明这些 Whittaker 函数可以与统计机械系统的配分函数等同。他们假设杨-巴克斯特方程是这些惠特克函数性质的基础。我们证实了这一点，并用量子仿射李超代数 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$ 的方程确定了相应的 Yang-Baxter 方程，通过德林菲尔德扭曲修改以引入高斯和。 （变形参数 $v$ 专用于剩余场基数的倒数。）对于超熔群的主级数表示，Whittaker 模型不是唯一的。标准交织算子的散射矩阵是矢量值的。对于简单的反射，它是由 Kazhdan 和 Patterson 计算的，他们将其应用于广义 theta 级数。我们将证明简单反射的 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。
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.