Langlands Correspondence, L-functions and Automorphic Forms

局部阿廷根数的存在性由朗兰兹和德利涅（在某些情况下更早由Dwork）建立，它们是建立局部域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）。还有一些其他的例子，这些对象被定义为GL（n，F）的表示，例如GL（n，C）的外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方法的外立方体。作为本提案的第一个主题，研究者将研究一种稳健的技术，该技术可用于通过对应关系证明Weil组定义的这些因素的相等性。所涉及的技术是一个变形的论点，以及广义Shalika胚芽扩展贝塞尔函数的雅克和叶似乎是适合推广到其他群体。他还将使用亚瑟的结果在他即将出版的书，以解决某些问题有关朗兰兹包附加到一个亚瑟包，以及某些算术问题（外尔定律）的经典群体，他们的概括一般自旋群体。计算残留的交织运营商的经典群体的内窥镜方面，他一直在追求合作多年，也应该受益于亚瑟的性格indentities，他将探讨作为这个项目的一部分。他还将研究某些代表性理论的后果functorality。 接下来，他将继续通过Langlands-Shahidi方法研究p-adic L-函数的联合工作，并追求Langlands关于Beyond Endoscopy和Reciprocity的新想法，以及该方法可能推广到循环群和覆盖群。该提案涉及研究生和博士后的培训，并包括他们的具体问题。调查人员预计几个新的学生加入他和其他成员的数论组在普渡大学，并参与教学高水平的课程（例如，p-adic L-函数，自守形式，真实的李群的表示理论）来训练它们。理论的阿廷L-功能及其连接的互惠法（对应）朗兰兹是一个最美丽的部分数论的调查希望可以研究的学生不同水平的不同研讨会。在另一个层面上，他参与组织会议，并在几个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民族招聘，目前在该部的毕业生招聘委员会任职，重点是招聘妇女和少数民族学生。

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.