Skip to main content
SpringerLink
Log in

On the local factors of irreducible representations of quaternionic unitary groups

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper, we give a precise definition of the analytic \(\gamma \)-factor of irreducible representations of quaternionic unitary groups, which extends a work of Lapid–Rallis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arthur, J.: The endoscopic classification of representations. Orthogonal and symplectic groups. Amer. Math. Soc. Colloquium Publications, 61. Amer. Math. Soc., Providence, RI (2013)

  2. Branson, T., Ólafsson, G., Ørsted, B.: Spectral generating operators and intertwining operators for representations induced from a maximal parabolic subgroup. J. Funct. Anal. 135(1), 163–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Casselman, W.: The unramified principal series of \(p\)-adic groups. I. The spherical function. Compos. Math. 40(3), 387–406 (1980)

    MathSciNet  MATH  Google Scholar 

  4. Casselman, W., Osborne, M.S.: The restriction of admissible representations to \(\mathfrak{n}\). Mathematische Annalen 233(3), 193–198 (1978)

    MathSciNet  MATH  Google Scholar 

  5. Gan, W.T.: Doubling zeta integrals and local factors for metaplectic groups. Nagoya Math. J. 208, 67–95 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gan, W.T., Ichino, A.: Formal degrees and local theta correspondence. Invent. Math. 195(3), 509–672 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gelbert, S., Piatetski-Shapiro, I., Rallis, S.: Explicit constructions of automorphic \(L\)-functions. vol. 1254 of Lecture Notes in Mathematics, Springer-Verlag, Berlin (1987)

  8. Godement, R., Jacquet, H.: Zeta Functions of Simple Algebra. Lecture Notes in Mathematics, vol. 260. Springer-Verlag, Berlin, New York (1972)

    Book  MATH  Google Scholar 

  9. Henniart, G.La: conjecture de Langlands locale pour \({\rm GL}(3)\). Mém. Soc. Math. France (N.S.) 11–12, 186 (1984)

    MATH  Google Scholar 

  10. Igusa, J.: Some results on p-adic complex powers. Am. J. Math. 106, 1013–1032 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ikeda, T.: On the functional equation of the Siegel series. J. Number Theory 172, 44–62 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kaletha, T., Minguez, A., Shin, S W., White, P.: Endoscopic Classification of Representations: Inner Forms of Unitary Groups. arXiv:1409.3731 [math.NT]

  13. Karel, M.L.: Functional equations of Whittaker functions on \(p\)-adic groups. Am. J. Math. 101(6), 1303–1325 (1979)

    MathSciNet  MATH  Google Scholar 

  14. Knapp, A.W.: Local Langlands correspondence: the Archimedean case. In: Motives (Seattle, WA, 1991), vol. 55 of Proc. Sympos. Pure Math., pp. 393–410. Amer. Math. Soc., Providence, RI (1994)

  15. Kottwitz, R.: Sign changes in harmonic analysis on reductive groups. Trans. AMS 278, 289–297 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koike, K., Terada, I.: Young-diagrammatic methods for the representation theory of the groups \(\operatorname{Sp}\) and \(\operatorname{SO}\). In: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), vol. 47 of Proc. Sympos. Pure Math., pp. 437-447. Amer. Math. Soc., Providence, RI (1987)

  17. Lapid, E.M., and Rallis, S.: On the local factors of representations of classical groups. In: Automorphic Representations, \(L\)-functions and Applications: Progress and Prospects, vol. 11 of Ohio State Univ. Math. Res. Inst. Publ., pp. 309-359. de Gruyter, Berlin (2005)

  18. Mok, C.P.: Endoscopic classification of representations of quasi-split unitary groups. Mem. Amer. Math. Soc. 235 no. 1108 (2015)

  19. Piatetski-Shapiro, I., Rallis, S.: \(\epsilon \) factor of representations of classical groups. Proc. Nat. Acad. Sci. USA 83(13), 4589–4593 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Shimura, G.: Some exact formulas on quaternion unitary groups. J. Reine Angew. Math. 509, 67–102 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shin, S.W.: Automorphic Plancherel density theorem. Isr. J. Math 192(1), 83–120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tate, J. T.: Local constants. Prepared in collaboration with C. J. Bushnell and M. J. Taylor. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 89–131. Academic Press, London (1977)

  23. Wallach, N.R.: Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals. In: Representations of Lie groups, Kyoto, Hiroshima, 1986, vol. 14 of Adv. Stud. Pure Math., pp. 123–151. Academic Press, Boston, MA (1988)

  24. Yamana, S.: Degenerate principal series representations for quaternionic unitary groups. Israel J. Math. 185, 77–124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yamana, S.: The Siegel-Weil formula for unitary groups. Pacific. J. Math. 264(1), 235–256 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yamana, S.: \(L\)-functions and theta correspondence for classical groups. Invent. Math. 196(3), 651–732 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Yamana, S.: Siegel series for skew Hermitian forms over quaternion algebras. Abh. Math. Semin. Univ. Hambg. 87(1), 43–59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank my supervisor A. Ichino for many advices. The author also would like to thank the referee for sincere and useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Hirotaka Kakuhama

Authors
  1. Hirotaka Kakuhama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hirotaka Kakuhama.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kakuhama, H. On the local factors of irreducible representations of quaternionic unitary groups. manuscripta math. 163, 57–86 (2020). https://doi.org/10.1007/s00229-019-01153-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-019-01153-6

Mathematics Subject Classification

Search

Navigation