Topics in Automorphic Forms

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Fourier analysis implies that functions that are periodic -- for example, quantities depending on time that repeat their previous value a fixed amount of time later -- can be realized as sums of trigonometric functions. Functions with more complicated, non-commutative, periodicities are at the heart of modern number theory. A fundamental vision of Langlands predicts connections between such functions and symmetries related to arithmetic, that is, coming from roots of polynomial equations. This proposal is concerned with periodic functions in a new guise, when the functions are defined not on a group but on a finite cover of a group. This blends arithmetic and analysis in a new way. Moreover, the techniques to be studied may have connections to constructions in mathematical physics.The main objects of study in this project are automorphic forms on covering groups. The principal investigator will focus first on Eisenstein series, which are obtained by an averaging process. Many of the standard tools from automorphic forms do not carry over (e.g. Whittaker functionals are typically not unique), but a surprising picture blending representation theory (canonical bases, Mirkovi\'c-Vilonen cycles) and number theory is emerging. The principal investigator will investigate these and and their number-theoretic applications. He also will study the residues of metaplectic Eisenstein series, "higher theta series." There is much to be done to understand the unipotent orbits attached to these objects, to develop relations between such series on different groups, and to use them both globally and in local constructions. A third project concerns unique functionals and Iwahori Hecke algebras. It offers the potential for new constructions of unique functionals.