Metaplectic Eisenstein series, crystal graphs, and quantum groups

朗兰兹程序是现代数论的基础。这个程序描述了一组欧拉l函数，每个函数都附属于约化群G的一个自同构表示和G的l群的一个有限维复解析表示。朗兰兹通过研究爱森斯坦级数的常数系数和惠特克系数得出了关于这些l函数的猜想，因为这些系数可以用这样的l函数来表示。本文的重点是理解当自同构表示是约化群的幂点的元复盖上的自同构表示时所产生的狄利克雷级数。首席研究员和他的合作者最近的前期工作表明，即使在最简单的情况下——从博雷尔导出的爱森斯坦级数——元塑性爱森斯坦级数的惠特克系数也具有非常丰富的结构，并且与晶体图理论有关，晶体图理论也出现在量子群的研究中。对元塑性惠特克系数的研究有可能提供数论兴趣的一个丰富的新对象族。它也被提议进一步发展与量子群理论的联系。现代数论中的许多问题都具有局部到全局的性质：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些局部知识来做出全局陈述。例如，回到黎曼和狄利克雷，一个人在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，它的性质反映了所有的局部性质，它在s中的行为与开始的问题有关（通常以微妙的方式）。该方案旨在展示一类新的全局对象，也反映了局部到全局的原则，但以一种新的方式。在构建的序列中，当将素数放在一起构成一个全局对象时，不同的素数在取其乘积时相互作用，而不是独立组合。

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.