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, Volume 159, Issue 1–2, pp 117–159 | Cite as

On Shahidi local coefficients matrix

  • Dani SzpruchEmail author


In this article we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation \(\mathrm{I}(\sigma )\) of an n-fold cover of p-adic \(\mathrm{{SL}_{2}(\mathrm{F})}\) and an additive character \(\psi \). The conjugacy class of this matrix is an invariant of the inducing representation \(\sigma \) and \(\psi \) and its entries are linear combinations of Tate or Tate type \(\gamma \)-factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measures and use these to relate principal series representations of different coverings of \(\mathrm{{SL}_{2}(\mathrm{F})}\). While we do not assume that the residual characteristic of \(\mathrm{F}\) is relatively prime to n we do assume that n is not divisible by 4.

Mathematics Subject Classification



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We would like to thank Fan Gao for his remarks on earlier versions of the manuscript. We have essentially an equal role in writing Sect. 4.1. We would also like to thank Alexander Burstein, Francois Ramaroson, Sankar Sitaraman and Valentin Buciumas for useful discussions on the subject matter. Finally we would like to thank the referee for numerous suggestions which significantly improved the style and clarity of this paper. At the time this manuscript was prepared, the author was partially supported by a Simons Foundation Collaboration Grant 426446.


  1. 1.
    Alon, G.: Semicharacters of groups. Commun. Algebra 43(5), 1771–1783 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aritürk, H.: On the composition series of principal series representations of a three-fold covering group of \({\rm SL}(2,\, K)\). Nagoya Math. J. 77, 177–196 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banks, W.D.: Exceptional representations on the metaplectic group. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.), Stanford University (1994)Google Scholar
  4. 4.
    Brubaker, B., Buciumas, V., Bump, D.: A Yang–Baxter equation for metaplectic ice. arXiv preprint arXiv:1604.02206 (2016)
  5. 5.
    Chai, J.-S., Cong, X.-R.: A note on Weil index. Sci. China Ser. A 50(7), 951–956 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chinta, G., Offen, O.: A metaplectic Casselman–Shalika formula for \({\rm GL}_r\). Am. J. Math. 135(2), 403–441 (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Davydov, A.: Twisted automorphisms of group algebras. In: Noncommutative Structures in Mathematics and Physics. K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, pp 131–150 (2010)Google Scholar
  8. 8.
    Fesenko, I.B., Vostokov, S.V.: Local Fields and Their Extensions, 2nd edn., vol. 121 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, With a foreword by I. R. Shafarevich (2002)Google Scholar
  9. 9.
    Finkelberg, M., Lysenko, S.: Twisted geometric Satake equivalence. J. Inst. Math. Jussieu 9(4), 719–739 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gan, W.T., Gao, F.: The Langlands–Weissman program for Brylinski–Deligne extensions. To appear in Asterisque. arXiv preprint arXiv:1409.4039 (2014)
  11. 11.
    Gao, F.: Distinguished theta representations for certain covering groups. Pac. J. Math. 290(2), 333–379 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gao, F.: The Langlands–Shahidi L-functions for Brylinski–Deligne extensions. Am. J. Math. 140(1), 83–137 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gao, F., Shahidi, F., Szpruch, D.: On the local coefficients matrix for coverings of SL(2). To Appear in the Springer Proceedings Geometry, Algebra, Number Theory, and Their Information Technology Applications, a volume in honour of Kumar Murty’s 60th Birthday. Preprint Available at (2017)
  14. 14.
    Goldberg, D., Szpruch, D.: Plancherel measures for coverings of \(p\)-adic \(\text{ SL }_2(F)\). Int. J. Number Theory 12(7), 1907–1936 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Karimianpour, C.: The Stone–von Neumann Construction in Branching Rules and Minimal Degree Problems. PhD thesis, Université d’Ottawa/University of Ottawa (2016)Google Scholar
  16. 16.
    Kazhdan, D.A., Patterson, S.J.: Metaplectic forms. Inst. Hautes Études Sci. Publ. Math. 59, 35–142 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kubota, T.: On automorphic functions and the reciprocity law in a number field. In: Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2. Kinokuniya Book-Store Co., Ltd., Tokyo (1969)Google Scholar
  18. 18.
    Kudla, S.S.: Tate’s thesis. In: An Introduction to the Langlands Program (Jerusalem, 2001), pp. 109–131. Birkhäuser Boston, Boston (2003)Google Scholar
  19. 19.
    Lang, S.: Algebraic Number Theory, vol. 110 of Graduate Texts in Mathematics, 2nd edn. Springer-Verlag, New York (1994)Google Scholar
  20. 20.
    Langlands, R.P.: On the functional equations satisfied by Eisenstein series. In: Lecture Notes in Mathematics, vol. 544. Springer-Verlag, Berlin-New York (1976)Google Scholar
  21. 21.
    McNamara, P.J.: Principal series representations of metaplectic groups over local fields. In: Multiple Dirichlet series, L-Functions and Automorphic Forms vol. 300 of Progr. Math. pp. 299–327. Birkhäuser/Springer, New York (2012)Google Scholar
  22. 22.
    McNamara, P.J.: The metaplectic Casselman–Shalika formula. Trans. Am. Math. Soc. 368(4), 2913–2937 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ranga Rao, R.: On some explicit formulas in the theory of Weil representation. Pac. J. Math. 157(2), 335–371 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schmidt, R.: Some remarks on local newforms for \({\rm GL}(2)\). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Shahidi, F.: On certain \(L\)-functions. Am. J. Math. 103(2), 297–355 (1981)CrossRefzbMATHGoogle Scholar
  26. 26.
    Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups. Ann. Math. (2) 132 2, 273–330 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Shahidi, F.: Eisenstein series and automorphic \(L\)-functions. American Mathematical Society Colloquium Publications, vol. 58. American Mathematical Society, Providence, RI (2010)Google Scholar
  28. 28.
    Sweet, J.: Functional equations of \(p\)-adic zeta integrals and representations of the metaplectic group. preprint (1995)Google Scholar
  29. 29.
    Szpruch, D.: Computation of the local coefficients for principal series representations of the metaplectic double cover of \({\rm SL}_2({\mathbb{F}})\). J. Number Theory 129(9), 2180–2213 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Szpruch, D.: On the existence of a \(p\)-adic metaplectic Tate-type \({\tilde{\gamma }}\)-factor. Ramanujan J. 26(1), 45–53 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tang, S.: Principal series representations of metaplectic groups. arXiv preprint arXiv:1706.05145 (2017)
  32. 32.
    Tate, J.T.: Fourier analysis in number fields, and Hecke’s zeta-functions. In: Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, D.C., pp. 305–347 (1967)Google Scholar
  33. 33.
    Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Weil, A.: Basic Number Theory. Classics in Mathematics. Springer, Berlin (1995). Reprint of the second (1973) editionGoogle Scholar
  35. 35.
    Weil, A.: Fonction zêta et distributions. In Séminaire Bourbaki, Vol. 9. Soc. Math. France, Paris, Exp. No. 312, pp. 523–531 (1995)Google Scholar
  36. 36.
    Weissman, M.H.: Metaplectic tori over local fields. Pac. J. Math. 241(1), 169–200 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Weissman, M.H.: L-groups and parameters for covering groups. To appear in Asterisque. arXiv preprint arXiv:1507.01042 (2015)
  38. 38.
    Weissman, M.H.: Covers of tori over local and global fields. Am. J. Math. 138(6), 1533–1573 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceOpen University of IsraelRaananaIsrael

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