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Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters

  • Spencer LeslieEmail author
Article
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Abstract

Formulas for the product of an irreducible character \(\chi _\lambda \) of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal \({\mathcal {B}}(\lambda +\rho )\) go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of p-adic groups as a sum over the canonical basis \({\mathcal {B}}(-\infty )\), which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirković–Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce resonant Mirković–Vilonen polytopes as the corresponding geometric context. Focusing on the exceptional group \(G_2\), we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over \({\mathcal {B}}(\lambda +\rho )\) plus a geometric error term coming from finitely many crystals of resonant polytopes.

Keywords

p-adic reductive groups Spherical Whittaker functions Total positivity Crystal bases Mirković–Vilonen polytopes Resonance Tokuyama’s formula 

Mathematics Subject Classification

Primary 20G25 Secondary 22E50 33C15 

Notes

Acknowledgements

I want to thank Solomon Friedberg for introducing me to questions which led directly to this project, as well as for many helpful conversations. I also wish to thank both Ben Brubaker and Dan Bump for helpful conversations on crystal graphs, Whittaker functions, and other topics on multiple occasions. Finally, I wish to thank the anonymous referee who explained to me the notion of (upper) seminormality and suggesting the correct statement of Theorem 6.8.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

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