Invariant Distributions on p-adic Lie Algebras

p 进李代数上的不变分布

• 批准号: 0070649
• 负责人: Rebecca Herb
• 金额: $ 7.2万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070649&HistoricalAwards=false
• 关键词: Invariant; Distributions; adic; Lie; Algebras

Technical Description: Let G be a connected reductive p-adic group, and let L(G) be its Lie algebra. The proposal is to study the G-invariantdistributions on L(G) which are Fourier transforms of invariant distributions with compactly generated support. This class of distributions includes the Fourier transforms of orbital integrals, which are important for the theory of characters on the group G. Harish-Chandra proved that these distributions are given by functions which are locally constant on the set of regular elements of L(G), and can be normalized to be locally bounded on L(G). When G is a real Lie group, the restrictions of these functions to Cartan subalgebras have simple formulas because they satisfy differential equations. In the p-adic case, there are analogous formulas for the Fourier transforms of orbital integrals restricted to Cartan subalgebras of L(G), but they are only valid for large enough sufficiently regular elements in the Cartan subalgebra. These formulas can be used to develop a theory of the constant term analogous to that for the real case. One goal of this proposal is to use these formulas at infinity to prove global bounds for the restrictions of Fourier transforms of orbital integrals to Cartan subalgebras. In order to do this it is necessary to control behavior at infinity uniformly as regular elements approach singular hyperplanes. A further goal of this proposal is to suitably generalize this work to the class of distributions obtained as Fourier transforms of invariant distributions with compactly generated support. Non-technical Description: The theory of Fourier series and Fourier transforms was developed, starting in the 18th century, to study functions of a real variable. The idea is to write an arbitrary function as a sum or integral of the well-understood trigonometric functions. This theory today has many applications in the sciences, in engineering, and in mathematics. Many of the basic ideas involved in Fourier analysis can be extended to analyze functions on any space with sufficient symmetry. One class of spaces of special interest in physics and many areas of mathematics is linear algebraic groups and their Lie algebras, the analysis on the Lie algebra being a linearization of the analysis on the group. These groups and algebras can be realized as matrices. Classically, the entries of the matrices are real or complex numbers. However there is also an interesting theory when the entries come from other fields, in particular the fields of p-adic numbers. These fields are important in number theory, and the study of p-adic groups has many applications to number theory. In the classical situation, much of the analysis involves the use of differential equations. This tool is not available in the p-adic case. The goal of this proposal is to analyze the behavior at infinity of certain important distributions on p-adic Lie algebras. The classical results are available as motivation, but the techniques of proof are necessarily completely different.

技术说明：设G是一个连通的约化p-adic群，L（G）是它的李代数。 本文的目的是研究L（G）上的G-不变分布，它们是具有紧生成支集的不变分布的傅立叶变换。这类分布包括轨道积分的傅立叶变换，这对群G上的特征标理论很重要。 Harish-Chandra证明了这些分布是由L（G）的正则元集上的局部常数函数给出的，并且可以归一化为L（G）上的局部有界函数。 当G是真实的李群时，这些函数对Cartan子代数的限制由于满足微分方程而具有简单的公式。 在p-adic情形下，对于L（G）的Cartan子代数上的轨道积分的傅里叶变换也有类似的公式，但它们只对Cartan子代数中足够大的充分正则元素有效。 这些公式可以用来发展一个类似于真实的情况的常数项理论。这个建议的一个目标是使用这些公式在无穷远证明全球范围内的限制的轨道积分的傅立叶变换的嘉当子代数。为了做到这一点，有必要控制行为在无穷远一致的正规元素接近奇异超平面。这个建议的另一个目标是适当地推广这项工作的类的分布得到的傅立叶变换的不变分布与companimously生成的支持。非技术描述：傅立叶级数和傅立叶变换的理论是从世纪开始发展起来的，用来研究真实的变量的函数。 这个想法是写一个任意函数作为一个总和或积分的良好理解的三角函数。 这个理论今天在科学、工程和数学中有许多应用。 傅立叶分析中的许多基本思想可以扩展到分析任何具有足够对称性的空间上的函数。 一类空间特别感兴趣的物理和许多领域的数学是线性代数群和他们的李代数，分析的李代数是一个线性化的分析组。 这些群和代数可以被实现为矩阵。 传统上，矩阵的元素是真实的或复数。 然而，当条目来自其他领域，特别是p-adic数的领域时，也有一个有趣的理论。 这些领域在数论中很重要，并且p进群的研究在数论中有许多应用。 在经典的情况下，大部分的分析涉及到微分方程的使用。 该工具在p-adic情况下不可用。这个建议的目的是分析某些重要分布在p-adic李代数无穷远处的行为。 经典的结果可以作为动机，但证明的技术必然完全不同。