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Duality for Metaplectic Ice Bump, Daniel


This is a report on arXiv:1604.02206 and arXiv:1709.06500, joint with Brubaker, Buciumas and Gray. Whittaker functions on the $n$-fold metaplectic cover of $GL(r)$ over a nonarchimedean local field were studied by Kazhdan and Patterson, who computed the scattering matrix of the intertwining integrals on the Whittaker models. It was shown in 2016 by Brubaker, Buciumas and Bump that this scattering matrix coincides with the R-matrix of a quantum group, a twist of quantum affine $U_{\sqrt{q}}(\widehat{\mathfrak{gl}}(n))$, where $q$ is the residue cardinality. Moreover, they showed that the spherical Whittaker functions could be expressed as partition functions of solvable lattice models, whose internal structure is related to the quantum affine Lie superalgebra $U_{\sqrt{q}}(\widehat{\mathfrak{gl}}(n|1))$. In recent work, Brubaker, Buciumas, Bump and Gray proved that a second solvable lattice model has the same partition function using Yang-Baxter equations. There may be analogies in mathematical physics, such as Kramers-Wannier duality for the Ising model, where the high-temperature system and the low temperature system have essentially the same partition function.

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