Langlands Correspondence, L-functions and Automorphic Forms

局部Artin根数的存在是建立局部域F上Weil-Deligne群的n维连续表示与GL(n，F)的不可约容许表示之间的局部朗兰兹对应的关键对象。事实上，只有当Weil群表示的张量积的根数和L函数与两个GL(…，F)的对应表示的Rankin乘积因子定义的根数和张量积函数相等时，才能得到唯一的对应。还有一些其他实例，其中这些对象被定义为GL(n，F)的表示，例如GL(n，C)的外部正方形和对称正方形表示，以及利用朗兰兹-沙希迪方法当n小于或等于8时的外部立方体。作为本提案的第一个主题，研究人员将研究一种稳健的技术，该技术可以用来通过对应关系证明由Weil群定义的那些因素是相等的。所涉及的技巧是形变论证，以及Jacquet和Ye对Bessel函数的广义Shalika芽展开，这似乎适合于推广到其他群。他还将利用亚瑟在他即将出版的书中的结果来解决与附着在亚瑟分组上的朗兰兹分组有关的某些问题，以及经典群的某些算术问题(韦尔定律)，以及它们对一般自旋群的推广。在内窥镜检查方面计算经典基团的缠绕算子的残留量，他多年来一直在合作，这也应该受益于亚瑟的性格认同，他将作为这个项目的一部分进行探索。他还将研究功能主义的某些表象理论结果。接下来，他将继续他的共同工作，通过朗兰兹-沙希迪方法来研究p-标量L函数，并追求朗兰兹关于超越内窥镜和互易性的新想法，以及将该方法推广到环群和覆盖群的可能性。该提案涉及研究生和博士后的培养，并包括了他们面临的具体问题。这位研究人员希望有几个新学生加入他和普渡大学数论小组的其他成员的行列，并参与高级课程的教学(例如，p-进L函数、自同构形、实李群的表示理论)来培训他们。Artin L函数理论及其与朗兰兹对易律(对应)的联系是数论中最美的部分之一，研究者希望不同层次的学生可以在不同的研讨会上学习它。在另一个层面上，他参与组织会议，并在几家知名期刊的编辑委员会和小组中任职。此外，他仍然参与指导和少数族裔招聘，目前在该部毕业生招聘委员会任职，重点是招聘妇女和少数族裔学生。

Local Artin root numbers whose existence were established by Langlands and Deligne (in some cases earlier by Dwork) are crucial objects in establishing the local Langlands correspondence between n-dimensional continuous representations of the Weil-Deligne group over a local field F and irreducible admissible representations of GL(n,F). In fact, a unique correspondence is obtained only after the root numbers and L-functions attached to tensor products of representations of Weil group are shown to equal to those defined by Rankin product factors for corresponding representations of two GL(.,F). There are some other instances where these objects are defined for representaions of GL(n,F) such as exterior square and symmetric square representations of GL(n,C), as well exterior cube when n is less than or equal to 8 by means of the Langlands-Shahidi method. As the first topic in this proposal, the investigator will study a robust technique which can be used to prove the equality of these factors by those defined for Weil group through the correspondence. Techniques involved are a deformation argument as well as a generalized Shalika germ expansion for Bessel functions by Jacquet and Ye which seems to be amenable to generalization to other groups. He will also use Arthur's results in his upcoming book to resolve certain questions concerning the Langlands packet attached to an Arthur packet, as well as certain arithmetic questions (Weyl laws) for classical groups, and their generalizations to general spin groups. Computing the residues of intertwining operators for classical groups in terms of endoscopy which he has been pursuing in collaboration for many years, should also benefit from Arthur's character indentities which he will explore as part of this project. He will also study certain representation theoretic consequences of functoriality. Next he will continue his joint work on studying p-adic L-functions through the Langlands-Shahidi method, and pursue Langlands new ideas on Beyond Endoscopy and Reciprocity, as well as the possible generalization of the method to loop groups and covering groups.The proposal involves training of graduate students and postdocs and includes specific problems for them. The investigator expects several new students to join him and other members of the Number Theory group at Purdue and is involved in teaching high level courses (e.g.,p-adic L-functions, automorphic forms, representation theory of real Lie groups) to train them. Theory of Artin L-functions and its connection with reciprocity law (correspondence) of Langlands is one of the most beautiful parts of number theory which the investigator hopes can be studied by students of different level in different seminars. On another level, he is involved in organizing conferences as well as serving in editorial boards of several prominent journals as well as panels. Moreover, he remains involved in mentoring and minority hiring and currently serves on the Department's Graduate Recruitment Committee with emphasis on recruiting women and minority students.