Metaplectic ice

Ben Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg, Paul E. Gunnells

Research output: Chapter in Book/Report/Conference proceedingChapter

14 Scopus citations

Abstract

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 function was given in terms of Gelfand-Tsetlin patterns in (Brubaker et al., Ann. of Math. 173(2):1081-1120, 2011; McNamara, Duke Math. J. 156:29-31, 2011), and we translate this description into an expression of the values of the Whittaker function as partition functions of a six-vertex model. Properties of the Whittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action (Chinta and Gunnells, J. Amer. Math. Soc. 23:189-215, 2010; Chinta and Offen, Amer. J. Math., 2011).

Original languageEnglish (US)
Title of host publicationMultiple Dirichlet Series, L-functions and Automorphic Forms
PublisherBirkhauser Boston
Pages65-92
Number of pages28
ISBN (Electronic)9780817683344
ISBN (Print)9780817683337
DOIs
StatePublished - Jan 1 2012

Keywords

  • Crystal graph
  • Metaplectic group
  • Solvable lattice model
  • Weyl group multiple dirichlet series
  • Whittaker function
  • Yang-baxter equation

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