Metaplectic Whittaker functions and quantum groups

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Whittaker functions are important in both number theory and physics. In number theory, they arise in the Fourier expansions of automorphic forms. In physics, they appeared in Kostant's work on the Toda lattice and in recent work of Gerasimov, Lebedev and Oblezin with connections to statistical and quantum physics and mirror symmetry.Brubaker, Bump, Chinta, Friedberg and Gunnells have been investigating Whittaker functions on metaplectic groups.They found that these have expressions as sums over Kashiwara crystals. Moreover they have expressions as partition functions of statistical physics, in which the Boltzmann weights are Gauss sums. This allows the Yang-Baxter equation to be introduced as a new tool in their study and opens up a new field of investigation.The proposer has been working with Brubaker, Chinta, Friedberg and Gunnells over the last few years over a project which began in number theory but is developing connections with mathematical physics. The objects under study are "Dirichlet Series" that resemble the Riemann zeta function, which is fundamental in number theory. Their symmetries, known as "functional equations" resemble the symmetries of Lie groups. It has recently been found that they may be realized in terms of a class of models arising in statistical and quantum physics, which has opened up a new area of investigation.