Studies in Representation Theory

这一建议包含两个独立的线索，都与约化李群的表示理论有关：研究这类群的不可约么正表示，以及附在自同构表示上的L函数的函数方程。在与Kari Vilonen的合作中，Schmidd将M.Saito的混合Hodge模理论应用到不可约酉表示的研究中，目的是得到酉性的准则。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，并从自同构分布的观点出发，在其他方法无法获得的情况下，推导出函数方程。经典傅立叶分析是研究一个或多个实变量函数的绝对基本工具。在20世纪，傅立叶分析被扩展到交换群、紧群，最后扩展到非紧、非交换群。不可约么正表示构成了傅立叶分析的基本构件。虽然Harish-Chandra构造了足够多的不可约酉性表示来对约化李群进行傅里叶分析，但是所有的不可约酉化表示对于约化李群的商空间上的傅里叶分析是必要的。存在许多部分结果，但它们并不符合连贯的、总体的图景。施密德和维洛宁最近就一元性问题提出了一个意义深远的猜想。他们的目的是验证这一猜想，并探索其各种含义。Riemann的Zeta函数对素数的深层性质进行了编码，狄利克雷L函数对有理数的阿贝尔扩张中的素数也做了同样的事情。这些函数是正则的，除了某些众所周知的极点外，它们都有欧拉积，并且满足函数方程。推测朗兰兹L函数在一般数域中起着类似的作用。它们是用欧拉乘积来定义的，但从定义上看，它们的分析性质并不明显。在重要的特殊情况下，建立了函数方程和正则性。米勒和施密德预计，他们的方法将适用于几个新的案例，并显著简化一些现有的论点。约化群的不可约么正表示在朗兰兹程序和数论的其他领域中起着重要的作用。数学物理文献中充斥着具有各种特殊性质的不可约么正表示的例子。对么正表示进行分类的更系统的方法将有助于统一数学物理的这一方面。沃根和他的合作者正在研究一种计算机算法，该算法将确定是否有任何指定的表示是可单元化的。对于一些群体，例如E8，这样的算法将使现有的计算设施严重紧张。施密德和维洛宁的猜想直接关系到这个庞大的计算问题，因为它允许显著减少需要检查的案例数量。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一大批数学家有用，远远超出了L函数专家的范围。

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.