Unitary Representations of Reductive Groups

“约化群的么正表示”摘要。Kirillov和Kostant的余共轭轨道哲学预言，可约Lie群的大多数不可约酉表示应该由(抛物线或上同调)归纳法从(仍未定义的)酉性表示产生。每个还原群应该只有有限多个单幂表示，并且这些表示应该与余伴表示的(有限多个)幂零轨道有关。这项提议包括两个部分。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刻画一小部分酉性表示的性质，即每个酉性表示都可以通过归纳法得到。(集合将包括所有的幂等表示，也包括各种补级数。)第二个(部分与威廉·格雷厄姆联合)涉及到构造幂等表示的思想。五十多年前，盖尔方提出了抽象调和分析的纲领，这是研究对称数学问题的一种非常普遍的方法。这样的问题出现在物理学中，而且几乎出现在数学的每一个部分。最基本的例子之一是与音乐声音有关的。在这种情况下，对称性是通过时间传递的：根据出现的声音类型，任何两个时间都是无法区分的。Gelfand的调和分析相当于将声音分解成音叉产生的“纯音”(音叉随着时间的推移以一种极其简单的方式变化)。盖尔芬德表明，在存在更复杂的对称性的情况下，类似的分析也是可能的。纯音的作用是通过“不可约的么正表示”来实现的。这个项目继续了许多人在不可约么正表示上所做的工作。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，其形式是所谓的希尔伯特空间和酉算子。这些正是形成量子力学所需要的物体。在物理学中，量子力学系统通过一种被称为量子化的(不完全理解的)过程从牛顿系统产生。20世纪60年代基里洛夫和科斯坦特提出的一种观点是，不可约的么正表示也应该通过对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好理解的，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部分，因此不乏可以考虑的例子和可以尝试的想法。

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.