Small Representations and Applications

戈丹萨文是继续他的工作代数方面的分析，与应用数论。本研究的主要工具是极小表示法（由Kazhdan和Savin发现），它在处理朗兰兹图的某些方面是成功的。实际上，通过使用极小表示法，可以得到标准方法无法得到的朗兰兹图的几个例子。广义的分析，处理满足某些微分方程的函数。一般说来，微分方程的分析是一个很难的问题。例如，微分方程来自流体动力学是出了名的困难。一个可能的原因是缺乏对称性。另一方面，如果有大量的对称性可以处理，那么相应的分析就比较容易，因为可以使用代数工具。这项研究正好处理了存在大量对称性的情况。本研究的主要目的是构造Lie型群的小表示。粗略地说，小是指在一个相对较小的空间上可以将许多对称性联合收割机组合在一起。下面的例子可以说明小的代表性的重要性。最著名（也是最臭名昭著）的群之一是Reisz-FischerMonster。是的，这个群体是巨大的，因此它的名字，但规模不是主要的困难。主要的问题是缺乏小的表示。事实上，60个字母的排列组--没有人会把这个组称为"怪物"--实际上比里斯-费舍尔怪物更大。然而，这组排列可以用60 × 60矩阵来表示。另一方面，怪物需要使用（大约）2000乘2000的矩阵。我们希望，小表示将是一个富有成效的工具，可以用来回答一些相关的问题，在分析和数论。

Gordan Savin is continuing his work on algebraic aspects of analysis,with applications in number theory. The main tools of this research areminimal representations (discoveredby Kazhdan and Savin) which have been successful in dealing with certainaspects of Langlands conjectures.Indeed, several instances of Langlands conjectures, out ofreach of standard methods, can be obtained throughuse of minimal representations.Analysis, broadly defined, deals withfunctions satisfying certain differential equations. Analysis ofdifferential equations, in general, is a very hard problem. For example,differential equations coming from the dynamics of fluids are notoriouslydifficult. One possible reason for this is a lack of symmetries. On theother hand, if there are plenty of symmetries to work with, thenthe corresponding analysis is easier, since algebraic tools can be used.This research deals precisely with situationswhen a large group of symmetries is present. The main purposeof this research is to construct small representations ofgroups of Lie type. Smallness, roughly speaking, refers to the fact thatit is possible to combine together many symmetries acting on arelatively small space.Importance of having small representationscan be illustrated with the following example.One of the most famous (and most notorious) groups is the Reisz-FischerMonster. Yes, this group is huge and hence its name,but the size is not the main difficultyhere. The main problem is the lack of small representations.Indeed, the group of permutations of 60 letters - nobody would considerthis group a ``monster'' - is in fact bigger then the Riesz-FischerMonster. However, this group of permutations can be written down(i.e. represented) by means of60 by 60 matrices. The monster, on the other hand, requires use of(roughly) 2000 by 2000 matrices. It is our hope that the smallrepresentations will be a fruitful tool that can be used to answersome relevant questions in analysis and number theory.