Proceedings of the Japan Academy, Series A, Mathematical Sciences

The local zeta integrals for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$

Tadashi Miyazaki

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Abstract

In this article, for irreducible admissible infinite-dimensional representations $\Pi$ and $\Pi'$ of $\mathit{GL}(2,\mathbf{C})$, we show that the local $L$-factor $L(s,\Pi \times \Pi')$ can be expressed as some local zeta integral for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 1 (2018), 1-6.

Dates
First available in Project Euclid: 5 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.pja/1515121223

Digital Object Identifier
doi:10.3792/pjaa.94.1

Mathematical Reviews number (MathSciNet)
MR3743720

Zentralblatt MATH identifier
06902808

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F30: Fourier coefficients of automorphic forms 22E46: Semisimple Lie groups and their representations

Keywords
Whittaker functions automorphic forms zeta integrals

Citation

Miyazaki, Tadashi. The local zeta integrals for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 1, 1--6. doi:10.3792/pjaa.94.1. https://projecteuclid.org/euclid.pja/1515121223


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References

  • T. Bröcker and T. tom Dieck, Representations of compact Lie groups, translated from the German manuscript, corrected reprint of the 1985 translation, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1995.
  • D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997.
  • H. Jacquet and R. P. Langlands, Automorphic forms on $\mathrm{GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970.
  • H. Jacquet, Automorphic forms on $\mathrm{GL}(2)$. Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin, 1972.
  • A. A. Popa, Whittaker newforms for Archimedean representations, J. Number Theory 128 (2008), no. 6, 1637–1645.
  • A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series. Vol. 3, translated from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, 1990.
  • E. T. Whittaker and G. N. Watson, A course of modern analysis, reprint of the 4th (1927) ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  • S.-W. Zhang, Gross-Zagier formula for $\mathrm{GL}_{2}$, Asian J. Math. 5 (2001), no. 2, 183–290.