Mathematical Sciences: Automorphic Forms on GL(r) and the Metaplectic Groups

本研究将集中于元群的自同构形式理论。它将试图证明与n阶狄利克雷特征相关的L-函数在GL(n)的n叠盖上作为爱森斯坦级数的傅立叶系数出现。这个结果可以应用于解析数论，也可以应用于瓦尔德斯普格定理的适当推广。它将进一步尝试证明与SL(r)的n重覆盖上的自同构形式相关的欧拉积是与GL(n)的n重覆盖上的θ函数的形式的Rankin-Selberg卷积。此外，它将试图证明GL（r+1）的n次覆盖上的爱森斯坦级数的傅里叶系数是GL（n-1）的n次覆盖上的函数的形式的Rankin-Selberg卷积。此外，Bump将研究例外群G2上的自同构形式和L-函数。最后从广义Barnes引理和广义超几何级数的角度研究了GL（r, r）上非分枝Whittaker函数的高卷积。本研究是在自同构形式领域，这是数论的一个分支，其中数论函数被编码成复杂的分析函数，允许深入的分析工具来承担数论问题。这一思想在许多方面得到了推广，并已被证明是许多领域的基本工具。多变量泛化是这项研究的重点，而Bump是这个方向的领导者。这项研究的结果将证明是非常令人兴奋的。

This research will focus on the theory of automorphic forms on the metaplectic groups. It will attempt to show that L- functions associated with Dirichlet characters of order n occur as Fourier coefficients of Eisenstein series on the n-fold covers of GL(n). This result would have applications to analytic number theory and to the formulation of the proper generalization of Waldspurger's theorem. It will further attempt to show that the Euler product associated with an automorphic form on the n-fold cover of SL(r) is the Rankin-Selberg convolution of the form with a theta function on the n-fold cover of GL(n). Also it will attempt to show that the Fourier coefficients of an Eisenstein series on the n-fold cover of GL(r+1) are Rankin-Selberg convolutions of the form with theta functions on the n-fold cover of GL(n-1). Also Bump will investigate automorphic forms and L- functions on the exceptional group G2. Finally work will be done on the higher convolutions of nonramified Whittaker functions on GL(r,R) from the point of view of generalized Barnes' lemmas and generalized hypergeometric series. This research is in the area of automorphic forms, a branch of number theory wherein number theoretic functions are encoded into complex analytic functions allowing the deep tools of analysis to come to bear on the number theory problems. This idea is generalized in many ways and has proved to be a fundamental tool in many areas. Multivariable generalizations are the focus of this research and Bump is a leader in this direction. The results of this research will prove to be very exciting.