Metaplectic Eisenstein series, crystal graphs, and quantum groups

Metaplectic Eisenstein 系列、晶体图和量子群

• 批准号: 1001326
• 负责人: Solomon Friedberg
• 金额: $ 11.8万
• 依托单位: Boston College
• 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: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001326&HistoricalAwards=false
• 关键词: Metaplectic; Eisenstein; series; crystal; graphs

The Langlands program is fundamental to modern number theory. This program describes a family of Eulerian L-functions, each attached to an automorphic representation on the adelic points of a reductive group G and a finite dimensional complex analytic representation of the L-group of G. Langlands was led to his conjectures about these L-functions by the study of the constant and Whittaker coefficients of Eisenstein series, as these coefficients can be expressed in terms of such L-functions. This proposal focusses on understanding the Dirichlet series that arise when the automorphic representation is one on a metaplectic cover of the adelic points of a reductive group. The recent prior work of the principal investigator and his collaborators has shown that even in the simplest case -- Eisenstein series induced from the Borel -- the Whittaker coefficients of metaplectic Eisenstein series have a remarkably rich structure, and are related to the theory of crystal graphs, which also arise in the study of quantum groups. The investigation of metaplectic Whittaker coefficients has the potential to provide a rich new family of objects of number-theoretic interest. It is also proposed to develop further connections to the theory of quantum groups.Many problems in modern number theory are of a local-to-global nature: one first studies them separately for each prime p, and then uses this local knowledge at all the primes to make a global statement. For example, going back to Riemann and Dirichlet, one takes information at p and encodes it in a function of a variable s, and then multiplies these functions to get a new function of s whose properties reflect all local properties and whose behavior in s is related to the problem that one began with (often in subtle ways). This proposal seeks to exhibit a new class of global objects, also reflective of a local-to-global principle, but in a new way. In the series under construction, when the primes are put together to make a global object, the different primes interact as one takes their product, rather than combining independently.

Langlands计划是现代数字理论的基础。 This program describes a family of Eulerian L-functions, each attached to an automorphic representation on the adelic points of a reductive group G and a finite dimensional complex analytic representation of the L-group of G. Langlands was led to his conjectures about these L-functions by the study of the constant and Whittaker coefficients of Eisenstein series, as these coefficients can be expressed in terms of such L功能。 该提案的重点是理解当自动形式表示是还原群体的Adelic点的元容器覆盖物上时出现的Dirichlet系列。 首席研究员及其合作者的最新作品表明，即使在最简单的情况下 - 来自Borel的Eisenstein Series系列 - 惠特克（Whittaker）的惠特克（Whittaker）系数Eisenstein系列的结构非常丰富，并且与晶体图的理论相关，这在量子组的研究中也存在。 对元素惠特克系数的调查有可能提供丰富的数量理论对象家族。 还有人建议与量子群体的理论建立进一步的联系。现代数字理论中的许多问题具有局部到全球性质：第一个对每个Prime P分别研究它们，然后在所有素数中使用这些本地知识来做出全球陈述。 例如，回到Riemann和Dirichlet，一个人在p上获取信息并以变量S的函数进行编码，然后乘以这些功能以获得S的新功能，其属性反映了所有局部属性，其在s中的行为与一个人开始的问题相关（通常以微妙的方式）。 该提议旨在展示一类新的全球对象，也反映了局部到全球原则，但以新的方式。 在正在建设的系列中，当素数放在一起以制造一个全球对象时，不同的素数会随着他们的产品而不是独立组合。