Representation Theory for Reductive Groups Over Local Fields

调查员建议研究一些相互关联的主题与代表性理论和调和分析约化群在当地的领域。 有四个总的主题，其中一些被进一步划分。第一个主题涉及一个奇怪的新公式，涉及字符的离散系列表示的真实的群体在厄米特对称的情况下，一个公式，无疑将需要比较莱夫谢茨公式与阿瑟-塞尔伯格迹公式。 第二个主题涉及各种设置中的转移因子：内部形式，志村品种，扭曲转移因子的下降。 第三个主题是关于志村品种的不良减少。 目前，这一点的优先程度较低，但在未来三年内可能会发生变化。 第四个主题由五个不同的，但有关轨道积分的约化群在非阿基米德局部领域的问题。在较少的技术语言的调查建议研究一些主题，属于理论的自守形式，一个美丽的数学领域的关系，所有三个主要分支的数学：代数，分析和几何。 过去的几十年一直是令人兴奋的时期，工人在这方面，其中一个亮点是必不可少的使用自守形式在怀尔斯的壮观证明费马大定理。 未来几十年将有许多令人兴奋的发展，研究人员希望通过解决提案中提出的一些问题直接促进这些发展，并间接帮助培训这一技术要求高的领域的年轻工人。

The investigator proposes to study a number of interrelated topics having to do with representation theory and harmonic analysis for reductive groups over local fields. There are four general topics, some of which are divided up further. The first topic concerns a strange new formula involving characters of discrete series representations of real groups in the Hermitian symmetric case, a formula that will no doubt be needed in order to compare the Lefschetz formula with the Arthur-Selberg trace formula. The second topic concerns transfer factors in various settings: inner forms, Shimura varieties, descent for twisted transfer factors. The third topic concerns bad reduction of Shimura varieties. Currently this has lower priority but that could change in the course of the next three years. The fourth topic consists of five distinct but related questions about orbital integrals for reductive groups over non-archimedean local fields. In less technical language the investigator proposes to study a number of topics that belong to the theory of automorphic forms, a beautiful area of mathematics with ties to all three main branches of mathematics: algebra, analysis and geometry. The last several decades have been exciting times for workers in this area, one of the highlights being the essential use of automorphic forms in Wiles's spectacular proof of Fermat's Last Theorem. The coming decades promise many further exciting developments, and the investigator hopes to contribute to these directly, by solving some of the questions raised in the proposal, as well as indirectly, by helping to train young workers in this technically demanding field.