Unitary representations of reductive p-adic groups: an algorithm

表示论是对线性空间中对称性的研究。一个物体或一个物理系统的对称性可以被编码成各种代数结构，例如群，以及它们的“表示”(线性空间上的作用)。该理论中的典型问题是如何从最基本的成分建立这些动作，并研究这些“原子”，即不可约表示。这个建议涉及不可约(酉性)表示的分类，更确切地说，目的是设计一个有限算法来确定可约p-ady群(考虑系数在p-ady域上的可逆方阵)的所有不可约酉性表示。从历史的角度看，(非紧)半单群的酉表示的分类(“酉对偶”问题)是表示论中最重要的悬而未决的经典问题之一。这个问题的起源可以追溯到盖尔芬德从1930年的S开始的“抽象调和分析”计划和维格纳关于洛伦兹群在物理学中的表示的工作。在过去的10年里，在Adams，van Leeuwen，Trapa和Vogan的工作中出现了新的想法，他们提出了判定实可约群的可Unitar化问题的有效算法，Schmidd和Vilonen的工作中也出现了新的想法，他们给出了Unitarity的几何解释(根据D-模的Hodge理论)。对于自同构形式的应用，必须对约化p-进群的酉对偶有同样精确的理解。因此，这一建议在“朗兰兹对应”的框架下提出了一个相应的方案，用来表示p-进还原群的么正表示。“朗兰兹对应”是现代数学的核心猜想之一。虽然Vogan等人的算法有一个形式上的部分，可以很容易地转换到p-adi环境中，但核心问题是深层次的，需要不同的方法和新的想法。除了对表象理论中这一经典问题的解答感到满意之外，该算法还将发现朗兰兹方案的几何和算术方面之间的新联系，因此它可能会对表象理论和自构形式的研究产生革命性的影响。

Representation theory is the study of symmetries in linear spaces. The symmetries of an object or a physical system can be encoded into various algebraic structures, such as groups, together with their "representations" (actions on linear spaces). The typical questions in the theory are how these actions are built from the most basic constituents and to study these "atoms", i.e., the irreducible representations. This proposal concerns the classification of irreducible (unitary) representations.More precisely, the aim is to devise a finite algorithm for the determination of all irreducible unitary representations of reductive p-adic groups (think of the invertible square matrices with coefficients in the field of p-adic numbers). From a historical perspective, the classification of unitary representations of (noncompact) semisimple groups ("the unitary dual" problem) is one of the most important unsolved classical problem in representation theory. The origins of this question can be traced back to Gelfand's programme of "abstract harmonic analysis'' from the 1930's and to Wigner's work on the representations of the Lorentz group in Physics. In the last 10 years, new ideas have emerged in the work of Adams, van Leeuwen, Trapa, and Vogan who produced an effective algorithm for deciding the unitarisability question for real reductive groups, and in the work of Schmid and Vilonen who gave a geometric interpretation (in terms of Hodge theory for D-modules) of unitarisability. For applications to automorphic forms, it is imperative to have a similarly precise understanding of the unitary dual of reductive p-adic groups. Thus this proposal advances a corresponding programme for unitary representations of p-adic reductive groups in the framework of the "Langlands correspondence", which is a vast set of conjectures central to much of modern Mathematics. While there is a formal part of the algorithm by Vogan et al which can be easily translated to the p-adic setting, the core problems are deep and require different methods and new ideas. In addition to the satisfaction of having an answer to this classical question in representation theory, the algorithm will uncover new connections between the geometric and arithmetic sides of the Langlands programme and therefore it could have a transformative impact on research in representation theory and in automorphic forms.