Metaplectic Whittaker functions and quantum groups

惠特克函数在数论和物理学中都很重要。在数论中，它们出现在自同构形的傅里叶展开中。在物理学中，它们出现在Kostant关于Toda晶格的工作中，以及Gerasimov、Lebedev和Oblezin最近的工作中，与统计和量子物理以及镜像对称性有关。Brubaker、Bump、Chinta、Friedberg和Gunnells一直在研究亚普勒群上的惠特克函数。他们发现，这些函数在Kashiwara晶体上有求和的表达式。此外，它们还表示为统计物理中的配分函数，其中玻尔兹曼权重为高斯和。这使得杨-巴克斯特方程作为一种新的工具被引入到他们的研究中，并开辟了一个新的研究领域。在过去的几年里，提出者一直在与布鲁贝克、钦塔、弗里德伯格和冈内尔斯合作一个项目，该项目始于数论，但正在发展与数学物理的联系。所研究的对象是类似黎曼-泽塔函数的“狄利克莱级数”，后者是数论中的基础。它们的对称性，也就是所谓的“函数方程”，类似于李群的对称性。最近发现，它们可以通过统计和量子物理中出现的一类模型来实现，这开辟了一个新的研究领域。

Whittaker functions are important in both number theory and physics. In number theory, they arise in the Fourier expansions of automorphic forms. In physics, they appeared in Kostant's work on the Toda lattice and in recent work of Gerasimov, Lebedev and Oblezin with connections to statistical and quantum physics and mirror symmetry.Brubaker, Bump, Chinta, Friedberg and Gunnells have been investigating Whittaker functions on metaplectic groups.They found that these have expressions as sums over Kashiwara crystals. Moreover they have expressions as partition functions of statistical physics, in which the Boltzmann weights are Gauss sums. This allows the Yang-Baxter equation to be introduced as a new tool in their study and opens up a new field of investigation.The proposer has been working with Brubaker, Chinta, Friedberg and Gunnells over the last few years over a project which began in number theory but is developing connections with mathematical physics. The objects under study are "Dirichlet Series" that resemble the Riemann zeta function, which is fundamental in number theory. Their symmetries, known as "functional equations" resemble the symmetries of Lie groups. It has recently been found that they may be realized in terms of a class of models arising in statistical and quantum physics, which has opened up a new area of investigation.