Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Local Rankin-Selberg convolutions for $\mathrm{GL}_{n}$:
explicit conductor formula

Authors: Colin J. Bushnell, Guy M. Henniart and Philip C. Kutzko
Journal: J. Amer. Math. Soc. 11 (1998), 703-730
MSC (1991): Primary 22E50
MathSciNet review: 1606410
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $F$ be a non-Archimedean local field and $n_{1}$, $n_{2}$ positive integers. For $i=1,2$, let $G_{i}=\mathrm{GL}_{n_{i}}(F)$ and let $\pi _{i}$ be an irreducible supercuspidal representation of $G_{i}$. Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant $\varepsilon (\pi _{1}\times \pi _{2},s,\psi )$ to the $\pi _{i}$ and an additive character $\psi $ of $F$. This object is of central importance in the study of the local Langlands conjecture. It takes the form

\begin{equation*}\varepsilon (\pi _{1}\times \pi _{2},s,\psi ) = q^{-fs}\varepsilon (\pi _{1} \times \pi _{2},0,\psi ), \end{equation*}

where $f=f(\pi _{1}\times \pi _{2},\psi )$ is an integer. The irreducible supercuspidal representations of $G=\mathrm{GL}_{n}(F)$ have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of $G$. This paper gives an explicit formula for $f(\pi _{1} \times \pi _{2},\psi )$ in terms of the inducing data for the $\pi _{i}$. It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in $\mathrm{GL}(n)$, developed by Bushnell and Kutzko.

References [Enhancements On Off] (What's this?)

  • 1. J. N. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR 771671
  • 2. Colin J. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of 𝐺𝐿_{𝑁}, J. Reine Angew. Math. 375/376 (1987), 184–210. MR 882297, 10.1515/crll.1987.375-376.184
  • 3. C. J. Bushnell and G. Henniart, Local tame lifting for $\operatorname {GL}(N)$ I: simple characters, Publ. Math. IHES 83 (1996), 105-233. CMP 97:05
  • 4. -, An upper bound on conductors for pairs, J. Number Theory 65 (1997), 183-196. CMP 97:16
  • 5. Colin J. Bushnell and Philip C. Kutzko, The admissible dual of 𝐺𝐿(𝑁) via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. MR 1204652
  • 6. -, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc., to appear.
  • 7. -, Semisimple types in $\operatorname {GL}_{n}$ Preprint, King's College London (1997).
  • 8. W. Casselman, Introduction to the theory of admissible representations of $\mathfrak{p}$-adic reductive groups, Preprint, University of British Columbia, 1974.
  • 9. I. M. Gel′fand and D. A. Každan, Representations of the group 𝐺𝐿(𝑛,𝐾) where 𝐾 is a local field, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 73–74 (Russian). MR 0333080
  • 10. Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
  • 11. Harish-Chandra, Harmonic analysis on reductive 𝑝-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 167–192. MR 0340486
  • 12. Guy Henniart, Caractérisation de la correspondance de Langlands locale par les facteurs 𝜀 de paires, Invent. Math. 113 (1993), no. 2, 339–350 (French, with English and French summaries). MR 1228128, 10.1007/BF01244309
  • 13. H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), no. 2, 199–214 (French). MR 620708, 10.1007/BF01450798
  • 14. H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, 10.2307/2374264
  • 15. François Rodier, Whittaker models for admissible representations of reductive 𝑝-adic split groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 425–430. MR 0354942
  • 16. F. Sauvageot, Principe de densité pour les groupes réductifs, Compositio Math. 108 (1997), 151-184. CMP 98:01
  • 17. Freydoon Shahidi, On certain 𝐿-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, 10.2307/2374219
  • 18. Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for 𝐺𝐿(𝑛), Amer. J. Math. 106 (1984), no. 1, 67–111. MR 729755, 10.2307/2374430
  • 19. Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for 𝑝-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, 10.2307/1971524
  • 20. Allan J. Silberger, Introduction to harmonic analysis on reductive 𝑝-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 22E50

Retrieve articles in all journals with MSC (1991): 22E50

Additional Information

Colin J. Bushnell
Affiliation: Department of Mathematics, King’s College, Strand, London WC2R 2LS, United Kingdom

Guy M. Henniart
Affiliation: Département de Mathématiques, URA 752 du CNRS, Université de Paris-Sud, 91405 Orsay Cedex, France
Email: or

Philip C. Kutzko
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Keywords: Local field, local constant of pair
Received by editor(s): October 3, 1997
Received by editor(s) in revised form: March 2, 1998
Additional Notes: The first author thanks Université de Paris Sud for hospitality and support for a period during the preparation of this paper.
The research of the second author was partially supported by EU TMR-Network “Arithmetic Geometry and Automorphic Forms” and a joint CNRS, NSF research grant.
The research of the third author was partially supported by NSF grant DMS-9003213 and a joint CNRS, NSF research grant.
Article copyright: © Copyright 1998 American Mathematical Society