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Metaplectic Whittaker functions and quantum groups

Metaplectic Whittaker 函数和量子群

基本信息

批准号：
1001079
负责人：
Daniel Bump
金额：
$ 29.99万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2010
资助国家：
美国
起止时间：
2010-07-15 至 2016-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001079&HistoricalAwards=false
关键词：
Metaplectic Whittaker functions quantum groups

项目摘要

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.

惠特克函数在数论和物理学中都很重要。在数论中，它们出现在自守形式的傅立叶展开中。在物理学中，它们出现在Kostant关于户田格的工作中，以及Gerasimov，Lebedev和Oblezin最近与统计和量子物理学以及镜像对称性有关的工作中。Brubaker，Bump，Chinta，Friedberg和Gunnells一直在研究亚晶群上的Whittaker函数。他们发现这些函数在柏原晶体上有求和的表达式。此外，它们还具有统计物理配分函数的表达式，其中玻尔兹曼权重是高斯和。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，提出者一直与Brubaker，Chinta，Friedberg和Gunnells合作进行一个项目，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“狄利克雷级数”，类似于黎曼zeta函数，这是数论的基础。它们的对称性，被称为“函数方程”，类似于李群的对称性。最近发现，它们可以用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。