Unitary representations of reductive p-adic groups: an algorithm

表征理论是对线性空间对称性的研究。物体或物理系统的对称性可以编码成各种代数结构，例如群，以及它们的“表示”（在线性空间上的作用）。该理论的典型问题是如何从最基本的成分中构建这些行为，以及如何研究这些“原子”，即不可约表征。这个建议涉及不可约（酉）表示的分类。更准确地说，目标是设计一种有限算法，用于确定可约p进群的所有不可约酉表示（考虑p进数域中具有系数的可逆方阵）。从历史的角度来看，（非紧）半单群的酉表示的分类问题（酉对偶问题）是表示理论中最重要的未解经典问题之一。这个问题的起源可以追溯到20世纪30年代Gelfand的“抽象谐波分析”计划，以及Wigner在物理学中关于洛伦兹群表示的工作。在过去的10年里，Adams、van Leeuwen、Trapa和Vogan的工作中出现了新的想法，他们提出了一个有效的算法来决定实约化群的单一性问题，而Schmid和Vilonen的工作则给出了单一性的几何解释（根据d模的Hodge理论）。对于自同构形式的应用，必须对可约p进群的酉对偶有类似的精确理解。因此，这一建议在“朗兰兹对应”的框架中提出了p进约化群的酉表示的相应方案，“朗兰兹对应”是许多现代数学中心的一套庞大的猜想。虽然Vogan等人的算法有一个形式部分可以很容易地转化为p进设置，但核心问题是深刻的，需要不同的方法和新的想法。除了对表征理论中这个经典问题的答案感到满意之外，该算法还将揭示朗兰兹纲领的几何和算术方面之间的新联系，因此它可能对表征理论和自同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.