# A Yang–Baxter equation for metaplectic ice

@article{Brubaker2019AYE, title={A Yang–Baxter equation for metaplectic ice}, author={Ben Brubaker and Valentin Buciumas and Daniel Bump}, journal={Communications in Number Theory and Physics}, year={2019} }

We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $\GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the… Expand

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#### References

SHOWING 1-10 OF 57 REFERENCES

Quantum affine algebras and holonomic difference equations

- Mathematics
- 1992

AbstractWe derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the… Expand

Schur Polynomials and The Yang-Baxter Equation

- Mathematics, Physics
- 2009

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map $${g \mapsto R (g)}$$ from $${ \rm{GL}(2, \mathbb{C}) \times… Expand

Metaplectic Ice

- Mathematics, Physics
- 2010

We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker… Expand

Block-compatible metaplectic cocycles

- Mathematics
- 1999

has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G of G, a… Expand

Split metaplectic groups and their L-groups

- Mathematics
- 2011

We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When $\tilde G$ is a central extension of a split connected reductive group over a local field… Expand

A metaplectic Casselman-Shalika formula for GLr

- Mathematics
- 2013

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in… Expand

Hecke modules from metaplectic ice

- Mathematics
- 2017

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of p-adic groups and R-matrices for… Expand

Quantum Groups

- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions… Expand

Structure and Representations of the Quantum General Linear Supergroup

- Physics, Mathematics
- 1996

Abstract: The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq… Expand

Twisted Whittaker models for metaplectic groups

- Mathematics
- 2015

Let G be a reductive group (over an algebraically closed field) equipped with the metaplectic data. In this paper we study the corresponding twisted Whittaker category for G. We construct and study a… Expand