Unitary Representations of Reductive Groups

还原群的酉表示

• 批准号: 9721441
• 负责人: David Vogan
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9721441&HistoricalAwards=false
• 关键词: Unitary; Representations; Reductive; Groups

Abstract of "Unitary representations of reductive groups." The philosophy of coadjoint orbits of Kirillov and Kostant predicts that most irreducible unitary representations of reductive Lie groups should arise by (parabolic or cohomological) induction from the (still undefined) unipotent representations. Each reductive group should have just finitely many unipotent representations, and these should be related to the (finitely many) nilpotent orbits of the coadjoint representation. This proposal has two parts. The first (partly joint with Susana Salamanca-Riba) concerns a precise version of the Kirillov-Kostant prediction. The goal is to characterize nicely a small set of unitary representations with the property that every unitary representation can be obtained from it by induction. (The set will include all unipotent representations, and also various complementary series.) The second (partly joint with William Graham) concerns an idea for constructing unipotent representations. More than fifty years ago, I. M. Gelfand set forth a program of abstract harmonic analysis, a very general way to study mathematical problems with symmetry. Such problems appear in physics, and in almost every part of mathematics. One of the most fundamental examples arises in connection with musical sounds. In that case the symmetry is passage through time: any two times are indistinguishable in terms of what kinds of sounds can appear. Gelfand's harmonic analysis amounts to decomposing a sound into the "pure tones" produced by a tuning fork (which change in an extremely simple way with the passage of time). Gelfand showed that a similar analysis was possible in the presence of more complicated symmetry. The role of the pure tones is played by "irreducible unitary representations." This project continues work done by many people on irreducible unitary representations. The central idea is this. The formal definition of an irreducible unitary representation involves linear a lgebra and Euclidean geometry, in the form of what are called Hilbert spaces and unitary operators. These are exactly the objects needed to formulate quantum mechanics. In physics, quantum mechanical systems arise from Newtonian ones by an (imperfectly understood) process known as quantization. An idea going back to Kirillov and Kostant in the 1960s is that irreducible unitary representations should also arise by "quantization" of some simple "Newtonian" analogues. These Newtonian analogues are fairly well understood, but the problem of quantizing them is difficult. On the other hand, similar quantization problems appear in many parts of mathematics and physics, so there is no shortage of examples to consider and ideas to try.

还原群的酉表示（Unitary Representations of Reductive Groups）。" 基里洛夫和科斯坦特的余伴随轨道哲学预言，约化李群的大多数不可约酉表示应该通过（抛物或上同调）归纳从（尚未定义的）幂幺表示产生。 每一个约化群都应该有恰好1000个幂幺表示，并且这些表示应该与余伴随表示的（1000个）幂零轨道相关。 这项建议有两个部分。 第一个（部分与苏珊娜·萨拉曼卡-里巴联合）涉及基里洛夫-科斯坦特预测的精确版本。 我们的目标是很好地描述一个小的酉表示集，其性质是每个酉表示都可以通过归纳法从它获得。 (The集合将包括所有的幂幺表示，也包括各种互补级数。 第二个（部分与威廉·格雷厄姆联合）涉及构造幂幺表示的想法。 五十多年前，我。M. Gelfand提出了一个程序的抽象谐波分析，一个非常普遍的方式来研究数学问题的对称性。 这样的问题出现在物理学中，几乎出现在数学的每一部分。 最基本的例子之一与音乐声音有关。 在这种情况下，对称性是通过时间：任何两个时间都无法区分什么样的声音可以出现。 Gelfand的谐波分析相当于将声音分解为音叉产生的“纯音”（随着时间的推移以极其简单的方式变化）。 盖尔芬德表明，在更复杂的对称性存在的情况下，类似的分析是可能的。 纯音的作用是由“不可约的幺正表示”来扮演的。“这个项目继续了许多人在不可约酉表示方面所做的工作。 中心思想是这样的。 不可约酉表示的形式定义涉及线性代数和欧几里得几何，以所谓的希尔伯特空间和酉算子的形式。 这些正是建立量子力学所需要的对象。 在物理学中，量子力学系统是通过一个被称为量子化的（不完全理解的）过程从牛顿系统中产生的。 一个想法可以追溯到基里洛夫和科斯坦特在20世纪60年代是，不可约酉表示也应该出现的“量子化”的一些简单的“牛顿”类似物。 这些牛顿的类似物是相当好理解的，但量化它们的问题是困难的。 另一方面，类似的量子化问题出现在数学和物理的许多部分，因此不缺乏可以考虑的例子和可以尝试的想法。