Studies in Representation Theory

这个建议包含两个独立的部分，它们都与约化李群的表示理论有关：这些群的不可约酉表示的研究，以及附属于自同构表示的l -函数的泛函方程。Schmid将与Kari Vilonen合作，将M. Saito的混合Hodge模理论应用于不可约酉表示的研究；目标是推导出一致性的准则。施密德将继续与史蒂夫·米勒合作。他们将证明l函数的正则性，并从自同构分布的角度推导出其他方法无法得到的泛函方程。经典傅里叶分析是研究一个或几个实变量函数的绝对基本工具。在20世纪，傅里叶分析扩展到阿贝尔群、紧群，最后扩展到非紧、非阿贝尔群。不可约的酉表示构成了傅里叶分析的基本组成部分。虽然Harish-Chandra构造了足够多的不可约酉表示来进行可约李群的傅里叶分析，但所有的不可约酉表示都是可约李群商空间傅里叶分析所必需的。存在着许多不完全的结果，但它们并不符合一个连贯的总体图景。Schmid和Vilonen最近提出了一个影响深远的关于统一性问题的猜想。他们的目的是验证这一猜想，并探讨其各种含义。黎曼的ζ函数编码了素数的深层性质，狄利克雷l函数也编码了有理数的阿贝尔扩展中的素数。这些函数是正则的，除了某些已知的极点，有欧拉积，并且满足函数方程。从推测上讲，朗兰兹l函数对一般数域起着类似的作用。它们是用欧拉积来定义的，但是它们的解析性质从定义上看并不明显。在重要的特殊情况下建立了函数方程和正则性。米勒和施密德希望他们的方法能适用于几个新的案例，并大大简化一些现有的论点。约化群的不可约酉表示在朗兰兹规划和数论的其他领域中起着重要的作用。数学物理文献中有大量具有各种特殊性质的不可约酉表示的例子。一个更系统的方法来分类的单位表示将有助于统一这方面的数学物理。Vogan和他的合作者正在研究一种计算机算法，该算法可以确定任何特定的表示是否可以统一。对于某些组，例如E8，这样的算法将严重紧张现有的计算设施。Schmid和Vilonen的猜想对这个庞大的计算问题有直接的影响，因为它可以显著减少需要检查的情况的数量。l -函数在代数和解析数论中占有中心地位，现在也引起了密码学家的兴趣。这使得关于l函数的重要新结果对大量数学家有用，远远超出了l函数专家的圈子。

This proposal contains two separate strands, both connected to the representation theory of reductive Lie groups: the study of irreducible unitary representations of such groups, and functional equations of L-functions attached to automorphic representations. In collaboration with Kari Vilonen, Schmid shall apply M. Saito's theory of mixed Hodge modules to the study of irreducible unitary representations; the objective is to derive criteria for unitarity. Schmid will continue his collaboration with Steve Miller. They will prove the regularity of L-functions, and derive functional equations, from the point of view of automorphic distributions, in cases that are inaccessible by other methods. Classical Fourier analysis is an absolutely fundamental tool for the study of functions of one or several real variables. In the 20th century, Fourier analysis was extended to abelian groups, compact groups, and finally to non-compact, non-abelian groups. Irreducible unitary representations constitute the basic building blocks of Fourier analysis. Although Harish-Chandra constructed enough irreducible unitary representations carry out Fourier analysis on reductive Lie groups, all irreducible unitary representations are necessary for Fourier analysis on quotient spaces of reductive Lie groups. Many partial results exist, but they do not fit into a coherent, general picture. Schmid and Vilonen recently formulated a far-reaching conjecture on the unitarity problem. Their aim is to verify the conjecture and to explore its various implications. Riemann's zeta function encodes deep properties of prime numbers, and Dirichlet L-functions do the same for primes in abelian extensions of the rational numbers. These functions are regular, except for certain well understood poles, have Euler products, and satisfy functional equations. Conjecturally Langlands L-functions play an analogous role for general number fields. They are defined in terms of Euler products, but their analytic properties are not obvious from the de¯nition. Functional equations and regularity have been established in important special cases. Miller and Schmid expect their method to apply in several new cases, and also to simplify some existing arguments significantly. Irreducible unitary representations of reductive groups play an important role in the Langlands program and other areas of number theory. The mathematical physics literature abounds with examples of irreducible unitary representation having various special properties. A more systematic approach to the classification of unitary representations would help to unify this aspect of mathematical physics. Vogan and his collaborators are working on a computer algorithm which would determine whether any specified representation is unitarizable. For some groups, for example E8, such an algorithm will severely strain existing computing facilities. The conjecture of Schmid and Vilonen has a direct bearing on this massive computational problem, by allowing to cut down signi¯cantly on the number of cases that need to be checked. L-functions occupy a central place in algebraic and analytic number theory, and are now of interest also to cryptographers. This makes significant new results about L-functions useful to a large group of mathematicians, well beyond the community of experts on L-functions.