Analysis and Invariant Theory for Gelfand Pairs

建议：DMS-9970552主要调查人员：盖尔·拉特克利夫，查尔·本森摘要。本文研究了幂零李群的紧扩张问题。一种是说，当N上的可积K-不变函数在卷积下交换时，紧李群K在幂零李群N上的作用产生Gelfand对。研究人员将继续他们对这种Gelfand对的研究，特别是他们对球函数的伴随空间的研究，特别是当N是Heisenberg群时。具体目标包括进一步发展计算球函数及其特征值的算法，研究广义二项式系数的恒等式，描述Gelfand空间的拓扑，以及应用于Schwartz空间中关于壳的极小理想的问题。傅立叶分析技术可追溯到热传导问题的研究，是数学家武器库中最强大的工具之一。傅立叶分析中的一个重要结构是所谓的卷积。在经典背景下，卷积具有重要的交换性。也就是说，人们可以在不影响乘法结果的情况下重新排列乘积中的因素的顺序。然而，傅立叶分析有一个现代而深远的推广，就像它的祖先与物理学中的重要问题有关，在这个问题中，卷积往往不是可交换的。这种分析是在被称为“李群”的对象的框架内完成的，在这种分析中的许多问题中，非对易性非常自然地出现，无法避免，导致了巨大的技术困难。Gelfand对提供了一种在某些非对易环境中恢复对易性的简单性的装置。每个Gelfand对提供了一个具有足够对称性的环境，以确保卷积的交换性。这样的配对是为了纪念俄罗斯数学家I.M.吉尔芬德而命名的，自20世纪50年代以来一直被研究。在分析半单李群的非对易环境时，它们的有用性现在已经得到了很好的证实。在过去的十年里，已经证明了Gelfand偶也出现在另一种非对易环境中，那就是幂零李群。这项研究的目标是对这类新的Gelfand对进行系统的研究和数学分析，包括开发显式计算的算法。

Proposal: DMS-9970552Principal Investigators: Gail Ratcliff, Chal BensonAbstract. This research project concerns analysis on compact extensions of nilpotent Lie groups. One says that the action of a compact Lie group K on a nilpotent Lie group N yields a Gelfand pair when the integrable K-invariant functions on N commute under convolution. The investigators will continue their study of such Gelfand pairs, in particular their study of the associated space of spherical functions, especially when N is a Heisenberg group. Specific goals include further development of algorithms for computing spherical functions and their eigenvalues, study of identities for generalized binomial coefficients, a description of the topology of the Gelfand space, and applications to questions concerning hull minimal ideals in Schwartz spaces.The techniques of Fourier analysis, which traces its roots to the study of problems in heat conduction, are among the most powerful tools in the mathematician's arsenal. An important construction in Fourier analysis is the so-called convolution product. In the classical setting, convolution has the important property of commutativity. That is, one can rearrange the order of the factors in the product without affecting the result of the multiplication. There is, however, a modern and far-reaching generalization of Fourier analysis, like its ancestor linked to important problems in physics, in which convolution often fails to be commutative. In many problems in this type of analysis, which is done in the framework of objects known as "Lie groups," noncommutativity arises quite naturally and cannot be avoided, leading to great technical difficulties. Gelfand pairs provide a device to recover the simplicity of commutativity in certain noncommutative settings. Each Gelfand pair provides an environment with enough symmetry to ensure commutativity of the convolution product. Such pairs are named in honor of the Russian mathematician I. M. Gelfand and have been studied since the 1950s. Their usefulness is now well established in connection with analysis in the noncommutative setting of semi-simple Lie groups. In the past decade, it has been shown that Gelfand pairs also arise in another non-commutative setting, that of nilpotent Lie groups. The goals of this research concern the systematic study and mathematical analysis of this new class of Gelfand pairs, including the development of algorithms for explicit computations.