Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-08T17:11:21.201Z Has data issue: false hasContentIssue false
  • English
  • Français

Local L-Functions for Split Spinor Groups

Published online by Cambridge University Press:  20 November 2018

Mahdi Asgari
Mahdi Asgari*
Affiliation:
Department of Mathematics, The University of Michigan, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA, email: asgari@math.lsa.umich.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the local $L$-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for $\text{G}{{\text{L}}_{2}}\,\times \,\text{G}{{\text{L}}_{3}}$.

Keywords

11F70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 673 - 693
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Asgari, M., On the holomorphy of local Langlands L-functions. PhD thesis, Purdue University, August, 2000.Google Scholar
[2] Ban, D., Self-duality in the case of SO(2n, F). Glas. Mat. Ser. III 34(54)(1999), 187–196.Google Scholar
[3] Bernstein, I. and Zelevinsky, A., Induced representations of reductive p-adic groups I. Ann. Sci. Ećole Norm. Sup. 10 (1977), 441–472.Google Scholar
[4] Borel, A., Automorphic L-functions. In: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math. (Oregon State Univ., Corvallis, OR, 1977), Part 2, Amer. Math. Soc., Providence, RI, 1979, 27–61.Google Scholar
[5] Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups. Preprint.Google Scholar
[6] Casselman, W. and Shahidi, F., On irreducibility of standard modules for generic representations. Ann. Sci. Ećole Norm. Sup. 31 (1998), 561–589.Google Scholar
[7] Fulton, W. and Harris, J., Representation theory, A first course. Graduate Texts in Math. 129, Springer-Verlag, New York, 1991.Google Scholar
[8] Jantzen, C., Degenerate principal series for orthogonal groups. J. Reine Angew. Math. 441 (1993), 61–98.Google Scholar
[9] Jantzen, C., Degenerate principal series for symplectic groups. Mem. Amer. Math. Soc. (488) 102(1993), xiv + 111 pp.Google Scholar
[10] Kim, H., On local L-functions and normalized intertwining operators. Preprint, 2000.Google Scholar
[11] Kim, H. and Shahidi, F., Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 . C. R. Acad. Sci. Paris. Sér. I Math. 331 (2000), 599–604.Google Scholar
[12] Kim, H. and Shahidi, F., Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 . Preprint, 2000.Google Scholar
[13] Langlands, R. P., Euler products. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. Yale Mathematical Monographs 1, Yale University Press, New Haven, Conn., 1971.Google Scholar
[14] Moeglin, C., Normalisation des opérateurs d’entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques p-adiques. Ann. of Math. (2) 151 (2000), 817–847.Google Scholar
[15] Muić, G., Some results on square integrable representations; irreducibility of standard representations. Internat. Math. Res. Notices (14) 1998 (1998), 705–726.Google Scholar
[16] Shahidi, F., On certain L-functions. Amer. J. Math. 103 (1981), 297–355.Google Scholar
[17] Shahidi, F., Local coefficients and normalization of intertwining operators for GL(n). Compositio Math. 48 (1983), 271–295.Google Scholar
[18] Shahidi, F., On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. of Math. (2) 127 (1988), 547–584.Google Scholar
[19] Shahidi, F., On multiplicativity of local factors. In: Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), Weizmann, Jerusalem, 1990, 279–289.Google Scholar
[20] Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math. (2) 132 (1990), 273–330.Google Scholar
[21] Shahidi, F., Twisted endoscopy and reducibility of induced representations for p-adic groups. Duke Math. J. 66 (1992), 1–41.Google Scholar
[22] Shahidi, F., On non-vanishing of twisted symmetric and exterior square L-functions for GL(n). Olga Taussky-Todd: in memoriam, Pacific J. Math. Special Issue (1997), 311–322.Google Scholar
[23] Springer, T. A., Linear algebraic groups. 2nd edition, Birkhäuser, Boston, 1998.Google Scholar
[24] Tadić, M., Notes on representations of non-archimedean SL(n). Pacific J. Math. 152 (1992), 375–396.Google Scholar
[25] Tadić, M., Representations of p-adic symplectic groups. Compositio Math. 90 (1994), 123–181.Google Scholar
[26] Tadić, M., On regular square integrable representations of p-adic groups. Amer. J. Math. 120 (1998), 159–210.Google Scholar
[27] Tadić, M., Square integrable representations of classical p-adic groups corresponding to segments. Represent. Theory 3 (1999), 58–59 (electronic).Google Scholar
[28] Tate, J., Number theoretic background. In: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math. (Oregon State Univ., Corvallis, OR, 1977), Part 2, Amer. Math. Soc., Providence, RI, 1979, 3–26.Google Scholar
[29] Zelevinsky, A., Induced representations of reductive p-adic groups II. On irreducible representations of GL(n). Ann. Sci. E´ cole Norm. Sup. 13 (1980), 165–210.Google Scholar