Duke Mathematical Journal

Arithmetic invariants of discrete Langlands parameters

Benedict H. Gross and Mark Reeder

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The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of $p$-adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the $L$-group. In this article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter. We attach a rational function in $x$ with rational coefficients to each discrete parameter, which specializes at $x=q$, the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. The order of this rational function at $x=0$ is also an important invariant of the parameter—it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the $L$-group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this article.

Article information

Duke Math. J. Volume 154, Number 3 (2010), 431-508.

First available: 7 September 2010

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Mathematical Reviews number (MathSciNet)

Primary: 11S15: Ramification and extension theory 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]


Gross, Benedict H.; Reeder, Mark. Arithmetic invariants of discrete Langlands parameters. Duke Mathematical Journal 154 (2010), no. 3, 431--508. doi:10.1215/00127094-2010-043. http://projecteuclid.org/euclid.dmj/1283865310.

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  • J. D. Adler, Refined anisotropic ${K}$-types and supercuspidal representations, Pacific J. Math. 185 (1998), 1--32.
  • J. Arthur, A note on $L$-packets, Pure Appl. Math. Q. 2 (2006), 199--217.
  • 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.
  • —, ``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.
  • A. Borel and J.-P. Serre, Sur certains sous-groupes des groupes de Lie compacts, Comment. Math. Helv. 27 (1953), 128--139.
  • N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1--3, Elem. Math. (Berlin), Springer, Berlin, 2002.
  • —, Lie Groups and Lie Algebras, Chapters 4--6, Elem. Math. (Berlin), Springer, Berlin, 2002.
  • C. J. Bushnell and G. Henniart, The Local Langlands Conjecture for ${\rm GL}(2)$, Grundlehren Math. Wiss. 335, Springer, Berlin, 2006.
  • R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley Classics Lib., Wiley, Chichester, 1993.
  • S. Debacker and M. Reeder, Depth-zero supercuspidal $L$-packets and their stability, Ann. of Math. (2) 169 (2009), 795--901.
  • P. Deligne, Les constants des équations fonctionnelles des fonctions $L$, Lecture Notes in Math. 349 (1973), 501--597.
  • —, Les constantes locales de l'équation fonctionnelle de la fonction $L$ d'Artin d'une représentation orthogonale, Invent. Math. 35 (1976), 299--316.
  • —, ``Application de la formule des traces aux sommes trigonométriques'' in Cohomologie étale, Lecture Notes in Math. 569, Springer, Berlin, 1977, 168--232.
  • E. Frenkel and B. Gross, A rigid irregular connection on the projective line, Ann. of Math. (2) 107 (2009), 1469--1512.
  • B. H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), 287--313.
  • —, On the motive of $G$ and the principal homomorphism ${\rm SL}_2\to\hat G$, Asian J. Math. 1 (1997), 208--213.
  • —, Irreducible cuspidal representations with prescribed local behavior, to appear in Amer. J. Math., preprint, 2009.
  • B. H. Gross and W.T. Gan, Haar measure and the Artin conductor, Trans. Amer. Math. Soc. 351 (1999), 1691--1704.
  • B. H. Gross and D. Prasad, On the decomposition of a representation of ${\rm SO}_n$ when restricted to ${\rm SO}_{n-1}$, Canad. J. Math. 44 (1992), 974--1002.
  • —, On irreducible representations of ${\rm SO}\sb {2n+1}\times{\rm SO}\sb {2m}$, Canad. J. Math. 46 (1994), 930--950.
  • 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.
  • 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.
  • Harish-Chandra, Harmonic Analysis on Reductive $p$-adic Groups, Lecture Notes in Math. 162, Springer, Berlin, 1970.
  • 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.
  • J. Heinloth, N. B. Chau, and Z. Yun, Kloosterman sheaves for reductive groups, preprint.
  • G. Henniart, Une preuve simple des conjectures de Langlands pour ${\rm GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), 439--455.
  • 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. ${\!}$;
  • K. Hiraga and H. Saito, On $L$-packets for inner forms of ${\rm SL}_n$, preprint, 2008.
  • J.W. Jones and D.P. Roberts, Database of Local Fields, preprint.
  • V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
  • N. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Ann. of Math. Stud. 116, Princeton Univ. Press, Princeton, 1988.
  • J.-L. Kim, Supercuspidal representations: An exhaustion theorem, J. Amer. Math. Soc. 20 (2007), 273--320.
  • A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Math. Ser. 36, Princeton Univ. Press, Princeton, 1986.
  • B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973--1032.
  • B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753--809.
  • R. E. Kottwitz, Stable trace formula: Cuspidal tempered terms, Duke Math. J. 51 (1984), 611--650.
  • P. C. Kutzko, Mackey's theorem for nonunitary representations, Proc. Amer. Math. Soc. 64 (1977), 173--175.
  • 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.
  • —, Representations of abelian algebraic groups, special issue, Pacific J. Math. 1997, 231--250.
  • 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.
  • D. Prasad, On the self-dual representations of a $p$-adic group, Int. Math. Res. Not. IMRN 1999, no. 8, 443--452.
  • 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.
  • —, Supercuspidal $L$-packets of positive depth and twisted Coxeter elements, J. Reine Angew. Math. 620 (2008), 1--33.
  • —, Torsion automorphisms of simple Lie algebras, to appear in Enseign. Math., preprint, 2009.
  • D. E. Rohrlich, Elliptic curves and the Weil-Deligne group, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, 1994, 125--157.
  • J.-P. Serre, Cohomologie des groupes discrets, Ann. of Math. Stud. 70, Princeton Univ. Press, Princeton, 1971, 77--169.
  • —, Conducteurs d'Artin des caractères réels, Invent. Math. 14 (1971), 173--183.
  • —, ``Modular forms of weight one and Galois representations'' in Algebraic Number Fields (Durham, England, 1975), Acad. Press, London, 1977, 193--268.
  • —, Local Fields, Grad. Texts in Math. 67, Springer, Berlin, 1979.
  • —, Exemples de plongements des groupes $\mathbf{PSL}_2(\mathbf F_p)$ dans des groupes de Lie simples, Invent. Math. 124 (1996), 525--562.
  • —, Galois Cohomology, corrected reprint, Springer Monogr. Math., Springer, Berlin, 2002.
  • —, Complète réductibilité, Astérisque 299 (2005), 195--217., Séminaire Bourbaki 2003/2004, no. 932.
  • N. Spaltenstein, Classes unipotentes et sous groups de Borel, Lecture Notes in Math. 946, Springer, Berlin, 1982.
  • T. A. Springer, Regular elements in finite reflection groups, Invent. Math. 25 (1974), 159--198.
  • —, A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279--293.
  • —, ``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.
  • R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, 1968.
  • —, Torsion in reductive groups, Adv. Math. 15 (1975), 63--92.
  • 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.
  • J. G. Thompson, A conjugacy theorem for $E_8$, J. Algebra 38 (1976), 525--530.
  • 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.
  • D. A. Vogan, ``The local Langlands conjecture'' in Representation Theory of Groups and Algebras, Contemp. Math. 145, Amer. Math. Soc., Providence, 1993, 305--379.
  • J.-K. Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579--622.
  • —, On the motive and cohomology of a reductive group, preprint, 2006.
  • A. Weil, Exercices dyadiques, Invent. Math. 27 (1974), 1--22.