Unitary Representations of Reductive Groups

“约化群的酉表示”摘要。Kirillov和Kostant的共伴轨道哲学预言了可约李群的大多数不可约酉表示应该由（仍未定义的）单幂表示（抛物或上同调）归纳产生。每个约化群应该有有限多个单幂表示，并且这些单幂表示应该与协表示的（有限多个）幂零轨道相关。这个建议有两个部分。第一个（部分与Susana Salamanca-Riba合作）涉及Kirillov-Kostant预测的精确版本。目标是很好地描述一个小的酉表示集合，其性质是每个酉表示都可以通过归纳法得到。（集合将包括所有的单幂表示，以及各种互补级数。）第二种观点（部分与威廉·格雷厄姆（William Graham）共同）关注的是构建单性表征的想法。五十多年前，I. M. Gelfand提出了一个抽象调和分析的程序，这是一种研究对称性数学问题的非常一般的方法。这样的问题出现在物理学和数学的几乎每一个部分。其中一个最基本的例子与音乐声音有关。在这种情况下，对称性是时间的流逝：任何两个时间都是无法区分的，就什么类型的声音可以出现而言。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，它以一种极其简单的方式变化）。Gelfand表明，在更复杂的对称性存在的情况下，类似的分析也是可能的。纯音的作用是由“不可约的酉表示”来发挥的。这个项目延续了许多人关于不可约酉表示的工作。中心思想是这样的。不可约酉表示的正式定义涉及线性代数和欧几里得几何，其形式被称为希尔伯特空间和酉算子。这些正是表述量子力学所需要的物体。在物理学中，量子力学系统是由牛顿力学系统通过一个被称为量子化的（尚不完全理解的）过程产生的。追溯到20世纪60年代基里洛夫和科斯坦特的一个想法是，不可约的酉表示也应该通过一些简单的“牛顿”类似物的“量子化”而产生。这些牛顿的类似物是相当容易理解的，但量化它们的问题是困难的。另一方面，类似的量化问题出现在数学和物理的许多领域，因此不缺乏可以考虑的例子和尝试的想法。

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.