The Langlands program is fundamental to modern number theory. This program describes a family of Eulerian L-functions, each attached to an automorphic representation on the adelic points of a reductive group G and a finite dimensional complex analytic representation of the L-group of G. Langlands was led to his conjectures about these L-functions by the study of the constant and Whittaker coefficients of Eisenstein series, as these coefficients can be expressed in terms of such L-functions. This proposal focusses on understanding the Dirichlet series that arise when the automorphic representation is one on a metaplectic cover of the adelic points of a reductive group. The recent prior work of the principal investigator and his collaborators has shown that even in the simplest case -- Eisenstein series induced from the Borel -- the Whittaker coefficients of metaplectic Eisenstein series have a remarkably rich structure, and are related to the theory of crystal graphs, which also arise in the study of quantum groups. The investigation of metaplectic Whittaker coefficients has the potential to provide a rich new family of objects of number-theoretic interest. It is also proposed to develop further connections to the theory of quantum groups.
Many problems in modern number theory are of a local-to-global nature: one first studies them separately for each prime p, and then uses this local knowledge at all the primes to make a global statement. For example, going back to Riemann and Dirichlet, one takes information at p and encodes it in a function of a variable s, and then multiplies these functions to get a new function of s whose properties reflect all local properties and whose behavior in s is related to the problem that one began with (often in subtle ways). This proposal seeks to exhibit a new class of global objects, also reflective of a local-to-global principle, but in a new way. In the series under construction, when the primes are put together to make a global object, the different primes interact as one takes their product, rather than combining independently.
This project has established new links between number theory and mathematical physics, links which also involve representation theory and combinatorics. On the number theory side, the objects of study are the Whittaker coefficients of Eisenstein series. A Whittaker coefficient is a generalization of a Fourier coefficient. An Eisenstein series is a function with certain invariance properties created by an averaging process. A fundamental theme of modern number theory has been that the Whittaker coefficients of certain Eisenstein series, attached to the adelic points of reductive groups, may be expressed in terms of number theoretic objects---Langlands L-functions---and so encode important arithmetic information. Moreover this connection may be used to study the L-functions that arise and to suggest conjectures for more general L-functions. On the mathematical physics and representation theory side, the objects of study are crystal graphs, which are related to representations of quantum groups. Quantum groups were introduced in order to better understand the Yang-Baxter equation, a key method used to exactly solve lattice models in statistical mechanics. Further links to lattice models have also been established. To establish these links, this project has investigated Eisenstein series where the group is replaced by a cover of the group. These covers arise naturally and arithmetically and so one may expect that the Whittaker coefficients of Eisenstein series on such groups will also be of number theoretic significance. The principal investigator and his collaborators Brubaker and Bump have established this in a number of general situations, showing that for classes of certain Eisenstein series (attached to the Borel subgroup) the coefficients may be expressed as infinite sums of Gauss sums and degenerate Gauss sums. The expressions, remarkably, involve crystal graphs. Moreover, they have shown that an alternative expession may be given using a new version of the 6-vertex model which appears in statistical mechanics. In this new version the Boltzmann weights are given by number-theoretic quantities (in contrast to the standard situation in mathematical physics, the new Boltzmann weights are not field free). For the one-fold cover, they have shown that the Yang-Baxter equation may be used to establish properties of the series, but for higher degree covers methods from number theory and polytope geometry appear to be required. These properties may be reinterpreted as a new instance of commuting transfer matrices. Additional work is in progress concerning other classes of Eisenstein series and their coefficients. The formulas for the Whittaker coefficients on covers are also linked, for the one-fold cover, to certain constructions in combinatorics, and it appears likely that this research will lead to some new results in this area. In addition, the principal investigator and his collaborators Chinta and Hoffstein have given new conjectures concerning the residues of these Eisenstein series, conjectures which they have verified for the rational function field and which it should be possible to test for higher degree function fields. This may help shed new light on the arithmetic significance of such residues. During this period of support, the principal investigator has given numerous lectures about this material, has co-organized a special semester on it at the Institute for Computational and Experimental Research in Mathematics (Spring 2013), and has organized several conferences focussing on mathematics related to this research. He has also co-advised one doctoral students, Ting-Fang Lee, who just defended her dissertation, which included work on multiple Dirichlet series over function fields, and has served as postdoctoral advisor to Lei Zhang, who is now working in the area.
