Langlands Correspondence, L-functions and Automorphic Forms

由Langlands和Deligne建立的局部Artin根数是建立局部域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Langlands对应关系的关键对象。实际上，只有当Weil群表示的张量积的根数和l -函数等于两个GL（.，F）对应表示的Rankin积因子定义的根数和l -函数时，才能得到唯一对应关系。在其他一些例子中，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，以及当n小于或等于8时，通过Langlands-Shahidi方法定义的外部立方体。作为本提案的第一个主题，研究者将研究一种鲁棒技术，该技术可用于证明这些因子的相等性，这些因子通过通信为Weil组定义。所涉及的技术是变形论证以及由Jacquet和Ye提出的Bessel函数的广义Shalika胚芽展开，它似乎可以推广到其他群。他还将在他即将出版的书中使用亚瑟的结果来解决有关附在亚瑟包上的朗兰兹包的某些问题，以及经典群的某些算术问题（Weyl定律），以及它们对一般自旋群的推广。在内窥镜中计算经典群的缠绕算子的余数这是他多年来一直在追求的合作，也应该受益于亚瑟的性格特征他将在这个项目中探索这一点。他还将研究功能性的某些表征理论结果。接下来他将继续通过Langlands- shahidi方法共同研究p进l函数，并追求Langlands关于Beyond Endoscopy和Reciprocity的新思想，以及将该方法推广到loop groups和covergroups的可能性。该提案涉及研究生和博士后的培训，并为他们提供了具体的问题。研究者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，并参与教授高级课程（例如，p进l函数，自同构形式，实李群的表示理论）来训练他们。Artin l -函数理论及其与朗兰兹互易律（对应律）的联系是数论中最美丽的部分之一，研究者希望不同层次的学生在不同的研讨会上对其进行研究。在另一个层面上，他参与组织会议，并在几家著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，目前在该部门的研究生招聘委员会任职，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.