Gamma factors and quadratic extension over finite fields

Article

Abstract

This paper characterizes \({\mathrm {GL}}_n({\mathbb {F}}_q )\)-distinguished cuspidal representations of \({\mathrm {GL}}_n({\mathbb {F}}_{q^2})\) in terms of the special values of their twisted gamma factors.

Mathematics Subject Classification

Primary 20C33 Secondary 11L05 

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References

  1. 1.
    Bump, D.: Notes on representations of \(\text{GL}(r)\) over a finite field. https://pdfs.semanticscholar.org/bca8/4c212fd6cf5feec34b8b0f3ebd224bff71c1.pdf
  2. 2.
    Chai, J.: Bessel functions and local converse conjecture of Jacquet. To appear in. J. Eur. Math. Soc. (2016)Google Scholar
  3. 3.
    Gelfand, S.I.: Representation of the general linear group over a finite field. In: Proceedings of Summer School of the Bolya-Janos Math. Soc., Budapest (1971) Halsted New York (1975)Google Scholar
  4. 4.
    Gelfand, S.I.: Representation of the full linear group over a finite field. Math. USSR Sb. 12(1), 13–39 (1970)CrossRefGoogle Scholar
  5. 5.
    Gelfand, I.M., Graev, M.I.: Construction of irreducible representations of simple algebraic groups over a finite field. Dokl. Akad. Nauk SSSR 147, 529–532 (1962). (Russian)MathSciNetGoogle Scholar
  6. 6.
    Gow, R.: Two multiplicity-free permutation representations of the general linear group \(\text{ GL }(n, q^2)\). Math. Z. 188(1), 45–54 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Green, J.A.: The characters of the finite general linear groups. Trans. Am. Math. Soc. 80, 402–447 (1955)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hakim, J., Hakim, J.: Distinguished \(p\)-adic representations. Duke Math. J. 62(1), 1–22 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hakim, J., Offen, O.: Distinguished representations of \(\text{ GL }(n)\) and local converse theorems. Manus. Math. 148(1–2), 1–27 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jacquet, H., Liu, Baiying: On the local converse theorem for p-adic \(\text{ GL }_n\). To appear in Amer. J. Math. (2016)Google Scholar
  11. 11.
    Jeffrey, A: Character tables for \(\text{ GL }(2), \text{ SL }(2), \text{ PGL }(2)\) and \(\text{ PSL }(2)\) over a finite field. http://www.math.umd.edu/~jda/characters/characters.pdf
  12. 12.
    Li, H-C.: A Note on Complex Representations of \(\text{ GL }(2, {\mathbb{F}}_q)\). http://math.ntnu.edu.tw/~li/note/REPGL2K.pdf
  13. 13.
    Nien, C.: A proof of finite field analogue of Jacquet’s conjecture. Amer J. Math. 136(3), 653–674 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Nien, C.: \(n\times 1\) local gamma factors and Gauss sums. Finite Fields Appl. 46, 255–270 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Offen, O.: On local root numbers and distinction. J. Reine Angew. Math. 652, 165–205 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Ok, Y.: Distinction and gamma Factors at 1/2: Supercuspidal Case. Ph.D. thesis, Columbia University. ProQuest LLC, Ann Arbor, MI (1997)Google Scholar
  17. 17.
    Piatetski-Shapiro, I.: Complex representations of \(\text{ GL }(2,K)\) for finite fields \(K\). Contemporary Mathematics, 16. American Mathematical Society, Providence, R.I. (1983)Google Scholar
  18. 18.
    Roditty, E-A.: On gamma factors and Bessel functions for representations of general linear groups over finite field. M.Sc. thesis, Tel-Aviv University (2010)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational Cheng Kung University and NCKUTainanTaiwan

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