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

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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.