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式的傅里叶展开式中。在物理学中，它们出现在Kostant关于Toda晶格的研究中，以及最近格拉西莫夫、列别捷夫和Oblezin与统计、量子物理和镜像对称的联系中。Brubaker、Bump、Chinta、Friedberg和gunnell一直在研究元塑群上的Whittaker函数。他们发现这些以柏原晶体的和表示。此外，它们的表达式是统计物理的配分函数，其中玻尔兹曼权重是高斯和。这使得Yang-Baxter方程作为一种新工具被引入他们的研究，并开辟了一个新的研究领域。在过去的几年里，这位提议者一直与布鲁贝克、钦塔、弗里德伯格和甘奈尔斯一起研究一个项目，该项目始于数论，但正在发展与数学物理的联系。研究的对象是类似于数论基础黎曼ζ函数的“狄利克雷级数”。它们的对称性，被称为“泛函方程”，类似于李群的对称性。最近发现，它们可以用统计物理学和量子物理学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.