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Formulas for tamely ramified supercuspidal characters of $\operatorname{GL}_3$

Author: Tetsuya Takahashi
Journal: Trans. Amer. Math. Soc. 355 (2003), 567-591
MSC (2000): Primary 22E50; Secondary 11F70
Published electronically: October 4, 2002
MathSciNet review: 1932714
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Abstract: Let $F$ denote a $p$-adic local field of residual characteristic $p\ne3$. This article gives formulas, valid on the regular elliptic set, for the irreducible supercuspidal characters of $\operatorname{GL}_3(F)$ which correspond to characters of a ramified Cartan subgroup. In the case in which $F$ does not contain cube roots of unity, i.e., the case in which ramified cubic extensions of degree $3$ over $F$ cannot be Galois, base change results concerning ``simple types" due to Bushnell and Henniart (1996) are used in the proofs.

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  • 1. J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Math. Studies 120, Princeton Univ. Press, 1989. MR 90m:22041
  • 2. A. Badulescu, Correspondance entre $\operatorname{GL}_n$ et ses formes intérieures en caractéristique positive, Thèse, Université de Paris-sud, Centre d'Orsay, 1999.
  • 3. C. J. Bushnell and A. Fröhlich, Gauss Sums and $p$-adic Division Algebras, Lecture Notes in Math. 987, Springer, Berlin, 1983. MR 84m:12017
  • 4. C. J. Bushnell and G. Henniart, Local Tame Lifting For GL(N) I: Simple characters, Inst. Hautes Études Sci. Publ. Math. No. 83 (1996), 105-233. MR 98m:11129
  • 5. H. Carayol, Représentations cuspidales du groupe linéaire Ann. Scient. École Norm. Sup. 17 (1984), 191-226. MR 86f:22019
  • 6. L. Corwin and R. Howe, Computing characters of tamely ramified $p$-adic division algebras, Pac. J. Math. 73 (1977), 461-477. MR 58:11238
  • 7. L. Corwin, A. Moy and P. J. Sally, Jr., Supercuspidal Character Formulas for $\operatorname{GL}_l$, Contemporary Mathematics 191, AMS, 1995, pp. 1-11. MR 96m:22037
  • 8. P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples $p$-adiques, in: Représentations des Groupes Réductifs sur un Corps Local, Herman, Paris, 1984, pp. 33-117. MR 86h:11044
  • 9. R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math. 260, Springer, Berlin, 1972. MR 49:7241
  • 10. G. Henniart, Correspondance de Jaquet-Langlands explicite I : le cas modéré de degré premier, in: Séminaire de Théorie des Nombres, Paris 1990-91, Progress in Math. 108, Birkhäuser, Basel, 1993, pp. 85-114. MR 95d:11064
  • 11. R. Howe, Kirillov theory for compact $p$-adic groups, Pac. J. Math. 73, (1977), 365-381. MR 58:28314
  • 12. -, Tamely ramified supercuspidal representations of $\operatorname{GL}_n(F)$, Pac. J. Math. 73 (1977), 437-460. MR 58:11241
  • 13. H. Hijikata, H. Saito, and M. Yamauchi, Representations of quaternion algebras over local fields and trace formula of Hecke operators, J. Number Theory 43 (1993), 123-167. MR 94c:11126
  • 14. H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Automorphic Forms on $\operatorname{GL}(3)$ II, Annals of Math. 109 (1979), 213-258. MR 80i:10034b
  • 15. P. Kutzko, Character formulas for supercuspidal representations of $\operatorname{GL}_l$, $l$ a prime, Amer. J. Math. 109 (1987), 201-222. MR 88k:22003
  • 16. A. Moy, Local constant and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863-930. MR 88b:11081
  • 17. F. Murnaghan, Asymptotic behaviour of supercuspidal characters of $p$-adic $\operatorname{GL}_3$ and $\operatorname{GL}_4$: The generic unramified case, Pacific J. Math. 148, (1991), 107-130. MR 92a:22023
  • 18. J. Rogawski, Representations of $\operatorname{GL}(n)$ and division algebras over $p$-adic field, Duke Math. J. 50 (1983), 161-196. MR 84j:12018
  • 19. T. Takahashi, Characters of cuspidal unramified series for central simple algebras of prime degree, J. Math. Kyoto Univ. 32-4 (1992), 873-888. MR 94e:11124
  • 20. -, Character formula for representations of local quaternion algebras (Wildly ramified case), J. Math. Kyoto Univ. 36-1 (1996), 151-197. MR 97f:11096
  • 21. -, On the irreducible very cuspidal representations II, J. Math. Kyoto Univ. 36-4 (1996), 889-910. MR 98k:22075
  • 22. J.-P. Serre, Linear Representations of Finite Groups, Springer Verlag, New-York, 1977. MR 56:8675

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Additional Information

Tetsuya Takahashi
Affiliation: Department of Mathematics and Information, College of Integrated Arts and Sciences, Osaka Prefecture University, 1-1 Gakuen-cho Sakai, 599-8531, Japan

Keywords: Characters, supercuspidal, base change
Received by editor(s): September 28, 1998
Received by editor(s) in revised form: May 2, 2002
Published electronically: October 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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