Metaplectic Iwahori Whittaker functions and supersymmetric lattice models
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arXiv
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2020
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| author | Brubaker, Ben Buciumas, Valentin Bump, Daniel Gustafsson, Henrik P. A. |
| author_facet | Brubaker, Ben Buciumas, Valentin Bump, Daniel Gustafsson, Henrik P. A. |
| contents | In this paper we compute new values of Iwahori Whittaker functions on $n$-fold metaplectic covers $\widetilde{G}$ of $\mathbf{G}(F)$ with $\mathbf{G}$ a split reductive group over a non-archimedean local field $F$. For every Iwahori Whittaker function $ϕ$, and for every $g\in\widetilde{G}$, we evaluate $ϕ(g)$ by recurrence relations over the Weyl group using novel "vector Demazure-Whittaker operators." The general formula and strategy of proof are inspired by ideas appearing in the theory of integrable systems. Specializing to the case of $\mathbf{G} = \mathbf{GL}_r$, we construct a solvable lattice model of a new type associated with the quantum affine super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ and prove that its partition function equals $ϕ(g)$. To prove this equality we match the recurrence relations on the lattice model side (obtained from the Yang-Baxter equation) to the recurrence relations for $ϕ(g)$ derived by using the representation theory of $\widetilde{G}$. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_15778 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Metaplectic Iwahori Whittaker functions and supersymmetric lattice models Brubaker, Ben Buciumas, Valentin Bump, Daniel Gustafsson, Henrik P. A. Representation Theory Number Theory Quantum Algebra 22E50, 82B23, 16T25, 05E05, 17B37, 11F70 In this paper we compute new values of Iwahori Whittaker functions on $n$-fold metaplectic covers $\widetilde{G}$ of $\mathbf{G}(F)$ with $\mathbf{G}$ a split reductive group over a non-archimedean local field $F$. For every Iwahori Whittaker function $ϕ$, and for every $g\in\widetilde{G}$, we evaluate $ϕ(g)$ by recurrence relations over the Weyl group using novel "vector Demazure-Whittaker operators." The general formula and strategy of proof are inspired by ideas appearing in the theory of integrable systems. Specializing to the case of $\mathbf{G} = \mathbf{GL}_r$, we construct a solvable lattice model of a new type associated with the quantum affine super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ and prove that its partition function equals $ϕ(g)$. To prove this equality we match the recurrence relations on the lattice model side (obtained from the Yang-Baxter equation) to the recurrence relations for $ϕ(g)$ derived by using the representation theory of $\widetilde{G}$. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions. |
| title | Metaplectic Iwahori Whittaker functions and supersymmetric lattice models |
| topic | Representation Theory Number Theory Quantum Algebra 22E50, 82B23, 16T25, 05E05, 17B37, 11F70 |
| url | https://arxiv.org/abs/2012.15778 |