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Correction factors for Kac–Moody groups and t-deformed root multiplicities

  • Dinakar MuthiahEmail author
  • Anna Puskás
  • Ian Whitehead
Article
  • 14 Downloads

Abstract

We study a correction factor for Kac–Moody root systems which arises in the theory of p-adic Kac–Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials \(m_\lambda \in \mathbb {Z}[t]\) indexed by positive imaginary roots \(\lambda \). At \(t=0\) these polynomials evaluate to the root multiplicities, so we consider \(m_\lambda \) to be a t-deformation of \({{\,\mathrm{mult}\,}}(\lambda )\). We generalize the Peterson algorithm and the Berman–Moody formula for root multiplicities to compute \(m_\lambda \). As a consequence we deduce fundamental properties of \(m_\lambda \).

Notes

Acknowledgements

We thank Alexander Braverman, Paul E. Gunnells, Kyu-Hwan Lee, Dongwen Liu, Peter McNamara, Manish Patnaik for helpful conversations. At the beginning of this project the first author was partially supported by a PIMS postdoctoral fellowship and the second author was supported through Manish Patnaik’s Subbarao Professorship in Number Theory and an NSERC Discovery Grant at the University of Alberta. The project started at the workshop “Whittaker functions: Number Theory, Geometry and Physics” at the Banff International Research Station in 2016; we thank the organizers of this workshop. We also thank the anonymous referee for helpful comments.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan
  2. 2.Department of Mathematics and StatisticsSwarthmore CollegeSwarthmoreUSA

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