×

zbMATH — the first resource for mathematics

Arithmetic invariants of discrete Langlands parameters. (English) Zbl 1207.11111
Let \(G\) be a connected reductive group over a nonarchimedean local field \(k\) of characteristic 0. Let \({\mathcal W}\) be the Weil group of \(k\). It is assumed that the connected centre of \(G\) is a \(k\)-anisotropic torus. Under this assumption a Langlands parameter \(\phi:{\mathcal W}\times \text{SL}_2({\mathbb C})\rightarrow {}^L G\) is discrete if and only if the centralizer in \(\hat G\) of the image of \(\phi\) is finite. The latter condition is equivalent to the condition that the adjoint \(L\)-function \(L(\phi, Ad, s)\) is regular at \(s = 0\). Thus for any discrete parameter \(\phi\) the \(\gamma\)-factor \(\gamma (\phi, Ad, s)\) is regular and nonzero at \(s = 0\). Put \(\gamma (\phi)=\gamma(\phi, Ad, 0)\). The authors show that, for any discrete parameter \(\phi\), there is a rational function \(\Gamma_{\phi}(X)\) with coefficients in \(\mathbb Q\) such that \(\gamma (\phi)/\gamma(\phi_0) = \Gamma_{\phi}(q)\), when the residue field of \(k\) has cardinality \(q\). Here \(\phi_0\) is the Steinberg parameter.
There is a conjecture, by Hiraga, Ichino and Ikeda, for the formal degree of a discrete series representation of \(G(k)\). This conjecture depends on the conjectural local Langlands correspondence. When reformulated using the Euler-Poincaré measure on \(G(k)\), the expression for the formal degree takes the form of a product of \(\Gamma_{\phi}(q)\) by a factor independent of \(q\).
The order of \(\Gamma_{\phi}(X)\) at \(X = 0\) is computed. It is conjectured that this order is always \(\geq0\), which amounts to an inequality for the Swan conductor of \(Ad\, \phi\). The conjecture is true for tamely ramified \(\phi\) and it is verified for some wildly ramified \(\phi\). Special attention is paid to the case where the inequality for the Swan conductor is an equality. In connection with the formal degree conjecture this leads to the definition and construction of ”simple” wild parameters and ”simple” supercuspidal representations of \(G(k)\). Assuming the formal degree conjecture (hence also the local Langlands conjecture), one can prove that the parameters of the simple supercuspidal representations are simple wild parameters.

MSC:
11S37 Langlands-Weil conjectures, nonabelian class field theory
11S15 Ramification and extension theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. D. Adler, Refined anisotropic \({K}\)-types and supercuspidal representations , Pacific J. Math. 185 (1998), 1–32. · Zbl 0924.22015 · doi:10.2140/pjm.1998.185.1 · nyjm.albany.edu:8000
[2] J. Arthur, A note on \(L\)-packets , Pure Appl. Math. Q. 2 (2006), 199–217. · Zbl 1158.22017 · doi:10.4310/PAMQ.2006.v2.n1.a9
[3] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup , Invent. Math. 35 (1976), 233–259. · Zbl 0334.22012 · doi:10.1007/BF01390139 · eudml:142406
[4] -, “Automorphic \(L\)-functions” in Automorphic Forms, Representations and \(L\)-functions, Proc. Symp. Pure Math. 33 (Corvallis, Ore., 1977), Amer. Math. Soc., Providence, 1979, 27–61.
[5] A. Borel and J.-P. Serre, Sur certains sous-groupes des groupes de Lie compacts , Comment. Math. Helv. 27 (1953), 128–139. · Zbl 0051.01902 · doi:10.1007/BF02564557 · eudml:139058
[6] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1–3 , Elem. Math. (Berlin), Springer, Berlin, 2002. · Zbl 0983.17001
[7] -, Lie Groups and Lie Algebras, Chapters 4–6 , Elem. Math. (Berlin), Springer, Berlin, 2002. · Zbl 1007.17003
[8] C. J. Bushnell and G. Henniart, The Local Langlands Conjecture for \({\mathrm GL}(2)\) , Grundlehren Math. Wiss. 335 , Springer, Berlin, 2006. · Zbl 1100.11041
[9] R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters , Wiley Classics Lib., Wiley, Chichester, 1993.
[10] S. Debacker and M. Reeder, Depth-zero supercuspidal \(L\)-packets and their stability , Ann. of Math. (2) 169 (2009), 795–901. · Zbl 1193.11111 · doi:10.4007/annals.2009.169.795 · annals.math.princeton.edu
[11] P. Deligne, Les constants des équations fonctionnelles des fonctions \(L\) , Lecture Notes in Math. 349 (1973), 501–597. · Zbl 0271.14011
[12] -, Les constantes locales de l’équation fonctionnelle de la fonction \(L\) d’Artin d’une représentation orthogonale , Invent. Math. 35 (1976), 299–316. · Zbl 0337.12012 · doi:10.1007/BF01390143 · eudml:142410
[13] -, “Application de la formule des traces aux sommes trigonométriques” in Cohomologie étale , Lecture Notes in Math. 569 , Springer, Berlin, 1977, 168–232.
[14] E. Frenkel and B. Gross, A rigid irregular connection on the projective line , Ann. of Math. (2) 107 (2009), 1469–1512. · Zbl 1209.14017 · doi:10.4007/annals.2009.170.1469 · annals.math.princeton.edu
[15] B. H. Gross, On the motive of a reductive group , Invent. Math. 130 (1997), 287–313. · Zbl 0904.11014 · doi:10.1007/s002220050186
[16] -, On the motive of \(G\) and the principal homomorphism \({\mathrm SL}_2\to\hat G\) , Asian J. Math. 1 (1997), 208–213. · Zbl 0942.20031
[17] -, Irreducible cuspidal representations with prescribed local behavior , to appear in Amer. J. Math., preprint, 2009.
[18] B. H. Gross and W.T. Gan, Haar measure and the Artin conductor , Trans. Amer. Math. Soc. 351 (1999), 1691–1704. JSTOR: · Zbl 0991.20033 · doi:10.1090/S0002-9947-99-02095-4 · links.jstor.org
[19] B. H. Gross and D. Prasad, On the decomposition of a representation of \({\mathrm SO}_n\) when restricted to \({\mathrm SO}_{n-1}\) , Canad. J. Math. 44 (1992), 974–1002. · Zbl 0787.22018 · doi:10.4153/CJM-1992-060-8
[20] -, On irreducible representations of \({\mathrm SO}_ {2n+1}\times{\mathrm SO}_ {2m}\) , Canad. J. Math. 46 (1994), 930–950. · Zbl 0829.22031
[21] B. H. Gross and M. Reeder, From Laplace to Langlands via representations of orthogonal groups , Bull. Amer. Math. Soc. (N.S.) 43 (2006), 163–205. · Zbl 1159.11047 · doi:10.1090/S0273-0979-06-01100-1
[22] B. H. Gross and N. Wallach, “Restriction of small discrete series representations to symmetric subgroups” in The Mathematical Legacy of Harish-Chandra (Baltimore, 1998) , Proc. Sympos. Pure Math. 68 , Amer. Math. Soc., Providence, 2000, 255–272. · Zbl 0960.22008
[23] Harish-Chandra, Harmonic Analysis on Reductive \(p\)-adic Groups , Lecture Notes in Math. 162 , Springer, Berlin, 1970. · Zbl 0202.41101
[24] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties , with an appendix by V. Berkovich, Ann. of Math. Stud. 151 , Princeton Univ. Press, Princeton, 2001. · Zbl 1036.11027
[25] J. Heinloth, N. B. Chau, and Z. Yun, Kloosterman sheaves for reductive groups , · Zbl 1272.14012 · arxiv.org
[26] G. Henniart, Une preuve simple des conjectures de Langlands pour \({\mathrm GL}(n)\) sur un corps \(p\)-adique , Invent. Math. 139 (2000), 439–455. · Zbl 1048.11092 · doi:10.1007/s002220050012
[27] K. Hiraga, A. Ichino, and T. Ikeda, Formal degrees and adjoint \(\gamma\)-factors , J. Amer. Math. Soc. 21 (2008), 283–304.; Correction , J. Amer. Math. Soc. 21 (2008), 1211–1213. \({\!}\); Mathematical Reviews (MathSciNet): · Zbl 1131.22014 · doi:10.1090/S0894-0347-07-00567-X · www.ams.org
[28] K. Hiraga and H. Saito, On \(L\)-packets for inner forms of \({\mathrm SL}_n\) , preprint, 2008.
[29] J.W. Jones and D.P. Roberts, Database of Local Fields , · Zbl 1140.11350 · arxiv.org
[30] V. G. Kac, Infinite-Dimensional Lie Algebras , 3rd ed., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0716.17022
[31] N. Katz, Gauss sums, Kloosterman sums, and monodromy groups , Ann. of Math. Stud. 116 , Princeton Univ. Press, Princeton, 1988. · Zbl 0675.14004
[32] J.-L. Kim, Supercuspidal representations: An exhaustion theorem , J. Amer. Math. Soc. 20 (2007), 273–320. · Zbl 1111.22015 · doi:10.1090/S0894-0347-06-00544-3
[33] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples , Princeton Math. Ser. 36 , Princeton Univ. Press, Princeton, 1986. · Zbl 0604.22001
[34] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group , Amer. J. Math. 81 (1959), 973–1032. JSTOR: · Zbl 0099.25603 · doi:10.2307/2372999 · links.jstor.org
[35] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces , Amer. J. Math. 93 (1971), 753–809. JSTOR: · Zbl 0224.22013 · doi:10.2307/2373470 · links.jstor.org
[36] R. E. Kottwitz, Stable trace formula: Cuspidal tempered terms , Duke Math. J. 51 (1984), 611–650. · Zbl 0576.22020 · doi:10.1215/S0012-7094-84-05129-9
[37] P. C. Kutzko, Mackey’s theorem for nonunitary representations , Proc. Amer. Math. Soc. 64 (1977), 173–175. · Zbl 0375.22005 · doi:10.2307/2041005
[38] R. P. Langlands, “On the classification of irreducible representations of real algebraic groups” in Representation Theory and Harmonic Analysis on Semisimple Lie Groups , Math. Surveys Monogr. 31 , Amer. Math. Soc., Providence, 1989, 101–170. · Zbl 0741.22009
[39] -, Representations of abelian algebraic groups , special issue, Pacific J. Math. 1997 , 231–250. · Zbl 0910.11045 · doi:10.2140/pjm.1997.181.231 · nyjm.albany.edu:8000
[40] G. Lusztig, “Leading coefficients of character values of Hecke algebras” in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) , Proc. Symp. Pure Math. 47 , Amer. Math. Soc., Providence, 1987, 235–262. · Zbl 0657.20037
[41] D. Prasad, On the self-dual representations of a \(p\)-adic group , Int. Math. Res. Not. IMRN 1999 , no. 8, 443–452. · Zbl 0923.22010 · doi:10.1155/S1073792899000227
[42] M. Reeder, Formal degrees and L-packets of unipotent discrete series representations of exceptional p-adic groups , with an appendix by Frank Lubeck, J. Reine Angew. Math. 520 (2000), 37–93. · Zbl 0947.20026 · doi:10.1515/crll.2000.023
[43] -, Supercuspidal \(L\)-packets of positive depth and twisted Coxeter elements , J. Reine Angew. Math. 620 (2008), 1–33. · Zbl 1153.22021 · doi:10.1515/CRELLE.2008.046
[44] -, Torsion automorphisms of simple Lie algebras , to appear in Enseign. Math., preprint, 2009.
[45] D. E. Rohrlich, Elliptic curves and the Weil-Deligne group , CRM Proc. Lecture Notes 4 , Amer. Math. Soc., Providence, 1994, 125–157. · Zbl 0852.14008
[46] J.-P. Serre, Cohomologie des groupes discrets , Ann. of Math. Stud. 70 , Princeton Univ. Press, Princeton, 1971, 77–169. · Zbl 0235.22020
[47] -, Conducteurs d’Artin des caractères réels , Invent. Math. 14 (1971), 173–183. · Zbl 0229.13006 · doi:10.1007/BF01418887 · eudml:142110
[48] -, “Modular forms of weight one and Galois representations” in Algebraic Number Fields (Durham, England, 1975), Acad. Press, London, 1977, 193–268. · Zbl 0366.10022
[49] -, Local Fields , Grad. Texts in Math. 67 , Springer, Berlin, 1979.
[50] -, Exemples de plongements des groupes \(\mathbf{PSL}_2(\mathbf F_p)\) dans des groupes de Lie simples , Invent. Math. 124 (1996), 525–562. · Zbl 0877.20033 · doi:10.1007/s002220050062
[51] -, Galois Cohomology , corrected reprint, Springer Monogr. Math., Springer, Berlin, 2002.
[52] -, Complète réductibilité , Astérisque 299 (2005), 195–217., Séminaire Bourbaki 2003/2004, no. 932. · Zbl 1156.20313
[53] N. Spaltenstein, Classes unipotentes et sous groups de Borel , Lecture Notes in Math. 946 , Springer, Berlin, 1982. · Zbl 0486.20025
[54] T. A. Springer, Regular elements in finite reflection groups , Invent. Math. 25 (1974), 159–198. · Zbl 0287.20043 · doi:10.1007/BF01390173 · eudml:142286
[55] -, A construction of representations of Weyl groups , Invent. Math. 44 (1978), 279–293. · Zbl 0376.17002 · doi:10.1007/BF01403165 · eudml:142532
[56] -, “Reductive groups” in Automorphic Forms, Representations, and \(L\)-functions (Corvalis, Ore., 1977) Part 1 , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 3–27.
[57] R. Steinberg, Endomorphisms of Linear Algebraic Groups , Mem. Amer. Math. Soc. 80 , Amer. Math. Soc., Providence, 1968. · Zbl 0164.02902
[58] -, Torsion in reductive groups , Adv. Math. 15 (1975), 63–92. · Zbl 0312.20026 · doi:10.1016/0001-8708(75)90125-5
[59] J. Tate, “Number Theoretic Background” in Automorphic Forms, Representations, and \(L\)-functions (Corvalis, Ore., 1977) Part 2 , Proc. Symp. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 2–26. · Zbl 0422.12007
[60] J. G. Thompson, A conjugacy theorem for \(E_8\) , J. Algebra 38 (1976), 525–530. · Zbl 0361.20027 · doi:10.1016/0021-8693(76)90235-0
[61] J. Tits, “Reductive groups over local fields” in Automorphic Forms, Representations, and \(L\)-functions (Corvalis, Ore., 1977) Part 1 , Proc. Symp. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 29–69. · Zbl 0415.20035
[62] D. A. Vogan, “The local Langlands conjecture” in Representation Theory of Groups and Algebras , Contemp. Math. 145 , Amer. Math. Soc., Providence, 1993, 305–379. · Zbl 0802.22005
[63] J.-K. Yu, Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622. JSTOR: · Zbl 0971.22012 · doi:10.1090/S0894-0347-01-00363-0 · links.jstor.org
[64] -, On the motive and cohomology of a reductive group , preprint, 2006.
[65] A. Weil, Exercices dyadiques , Invent. Math. 27 (1974), 1–22. · Zbl 0307.12017 · doi:10.1007/BF01389962 · eudml:142305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.