Mathematical Sciences: Smooth Representations of Reductive P-Adic Groups via G-Types

局部域上约化群的可容许表示已经成为一段时间以来研究的对象，利用朗兰兹程序将它们与局部数论和全局数论联系起来。在这些表征中，超尖形表征一直扮演着重要的角色。构造它们的一种方法是通过紧开子群的归纳法，这导致了通过将约化群限制到适当的紧开子群来研究可容许表示的哲学。Bushnell和Kutzko将这一理念成功地应用于一般或特殊线性群；这使得他们通过s类型的一般概念将其形式化。该方法还成功地研究了任意约化群的光滑表示，该约化群在某些旁子群的幂幂根下包含一个固定向量（“零能级情况”）。最后，初步结果表明，这种方法可以更普遍地适用于其他群体。PI将继续在这一领域进行调查，他将特别审议下列题目：(a)研究经典群体的s型；特别是经典群的可容许对偶的分类，(b) (a)在研究经典群的广义主级数的可约性及有关算法中的应用，(c)进一步研究零能级情况引起的问题；特别是可容许对偶在Langlands参数上的零能级部分的分类，以及零能级s型在可约性和r群研究中的应用。朗兰兹程序是数论的一部分。数论是对整数性质的研究，是数学中最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分中创造新的数学提供了动力。朗兰的程序是一种将数论与微积分联系起来的一般哲学；它体现了研究整数的现代方法。现代数论是非常技术性和深奥的，但它在理论计算机科学和编码理论等领域有着惊人的应用。

Morris 9623288 Admissible representations of reductive groups over local fields have been the object of study for some time by virtue of the Langlands program relating them to local and global number theory. Among these representations, the supercuspidal representations have always played a distinguished role as building blocks. One way to construct them is via induction from compact open subgroups, and this has led to the philosophy of studying admissible representations of a reductive group by restricting them to appropriate compact open subgroups. Bushnell and Kutzko have applied this philosophy with great success to the general or special linear groups; this has led them to formalize it via the very general notion of S-types. This approach has also been successful in studying the smooth representations of an arbitrary reductive group that contains a fixed vector under the prounipotent radical of some parahoric subgroup (the "level zero case"). Finally, preliminary results suggest this approach can work more generally for other groups. The PI will continue his investigations in this area, and in particular, he will consider the following topics: (a) The study of S-types for classical groups; in particular the classification of the admissible dual of classical groups, (b) Applications of (a) to the study of the reducibility of generalized principal series of classical groups and related arithmetic, (c) Further study of problems arising from the level zero case; in particular, the classification of the level zero part of the admissible dual in terms of Langlands parameters and applications of level zero S-types to the study of reducibility and R-groups. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. Th e Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.