Metaplectic Eisenstein series, crystal graphs, and quantum groups

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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.