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Twisted Poincaré series and zeta functions on finite quotients of buildings

Authors
  • Kang, Ming-Hsuan1
  • McCallum, Rupert2
  • 1 National Chiao-Tung University, Department of Applied Mathematics, Hsinchu, Taiwan , Hsinchu (Taiwan)
  • 2 University of Tübingen, Department of Mathematics, Tübingen, Germany , Tübingen (Germany)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Dec 08, 2018
Volume
49
Issue
3
Pages
309–336
Identifiers
DOI: 10.1007/s10801-018-0857-8
Source
Springer Nature
Keywords
License
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Abstract

In the case where G=SL2(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\hbox {SL}_{2}(F)$$\end{document} for a non-archimedean local field F and Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document} is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat–Tits tree of G by the action of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document}, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalized to other split simple algebraic groups of rank two over F and formulate a conjecture about how this might be generalized to groups of higher rank.

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