Multiple Dirichlet Series with Applications to Automorphic Representation Theory

In this project, Principal Investigator Brubaker and his collaborators seek to understand metaplectic forms, a generalization of automorphic forms to certain covers of split, reductive algebraic groups. Specifically, the proposed research focuses on constructing Dirichlet series in several complex variables, termed ``Multiple Dirichlet Series'' (MDS), with good analytic properties (e.g. functional equations and meromorphic continuation) and connecting these series to the Fourier-Whittaker coefficients of Eisenstein series on metaplectic groups. Since the construction of the metaplectic cover is intimately tied to reciprocity laws in number fields, the Dirichlet series and its polar residues contain families of automorphic forms twisted by characters built from power residue symbols. Analytic properties of the Dirichlet series then translate to arithmetic applications for automorphic forms, including non-vanishing results for automorphic L-functions. The construction and subsequent proof of analytic properties of these MDS uses new techniques in combinatorial representation theory, and another primary objective of this work is a deeper understanding of the connections between this representation theory and metaplectic forms. In the process, this structure is expected to illuminate further the relationship between combinatorial representation theory and the special case of Fourier-Whittaker coefficients of automorphic forms.

Historically, problems in number theory have centered around integer solutions to polynomial equations, so called ``Diophantine equations,'' which could be simply stated, but often extraordinarily hard to prove. It once appeared that these questions were the stuff of pure thought experiments, since the integers are discrete and should therefore have no bearing on the world and its continuous phenomena. But in the late 1960's, Robert Langlands developed a series of far-reaching conjectures known today as the Langlands' Program to investigate connections among number theory, arithmetic geometry and harmonic analysis; in fact, his conjectures were based on calculations involving a special case of the highly symmetric functions known as Eisenstein series, which are the principal objects of study in this proposal. The reach of Langlands' conjectures has been greatly expanded in the last several decades and now extends from methods for solving Diophantine equations to geometric versions with intimate connections to quantum field theory

and string theory, which attempt to explain the origins and expansion of our universe via a uniform treatment of fundamental forces including gravity and electromagnetism. That is, motivated by natural questions about solutions of polynomial equations in the integers, one obtains a new interpretation for profoundly important physical phenomena; in trying to answer discrete problems, one finds explanations of the continuous world. This project attempts to bring together a previously disparate community of researchers and students in number theory, automorphic forms, Lie groups, and combinatorics to further investigate analogous connections by studying large classes of more general Eisenstein series and their relations to the aforementioned disciplines.