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.

朗兰兹纲领是现代数论的基础。 这个程序描述了一族欧拉L-函数，每个函数都依附于约化群G的顶点上的一个自守表示和G的L-群的一个有限维复解析表示。朗兰兹是导致他的progratures关于这些L-职能的研究常数和惠特克系数的爱森斯坦系列，因为这些系数可以表示在这样的L-职能。 这个建议的重点是理解的狄利克雷级数时出现的自守表示是一个元复盖的adelic点的一个约化群。 主要研究者和他的合作者最近的先前工作表明，即使在最简单的情况下-从Borel诱导的Eisenstein系列-元Eisenstein系列的惠特克系数具有非常丰富的结构，并且与晶体图理论有关，这也出现在量子群的研究中。 调查metaplectic惠特克系数有可能提供一个丰富的新家庭的对象数论的兴趣。 现代数论中的许多问题都具有局部到整体的性质：人们首先对每个素数p分别研究它们，然后在所有素数上使用这种局部知识来做出整体陈述。 例如，回到黎曼和狄利克雷，人们在p处获取信息，并将其编码为变量s的函数，然后将这些函数相乘，得到一个新的s函数，其性质反映了所有局部性质，并且其在s中的行为与我们开始的问题有关（通常以微妙的方式）。 这个提议试图展示一种新的全局对象，也反映了局部到全局的原则，但以一种新的方式。 在正在构建的系列中，当素数被放在一起构成一个全局对象时，不同的素数会相互作用，而不是独立组合。