Algebra & Number Theory

Jordan blocks of cuspidal representations of symplectic groups

Corinne Blondel, Guy Henniart, and Shaun Stevens

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Abstract

Let G be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Mœglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a ramification theorem for G, giving a bijection between the set of endoparameters for G and the set of restrictions to wild inertia of discrete Langlands parameters for G, compatible with the local Langlands correspondence. The main tool consists in analyzing the Hecke algebra of a good cover, in the sense of Bushnell–Kutzko, for parabolic induction from a cuspidal representation of G× GLn, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine reducibility points; a criterion of Mœglin then relates this to Langlands parameters.

Article information

Source
Algebra Number Theory, Volume 12, Number 10 (2018), 2327-2386.

Dates
Received: 30 May 2017
Revised: 2 June 2018
Accepted: 20 July 2018
First available in Project Euclid: 14 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1550113225

Digital Object Identifier
doi:10.2140/ant.2018.12.2327

Mathematical Reviews number (MathSciNet)
MR3911133

Zentralblatt MATH identifier
07026820

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Keywords
local Langlands correspondence symplectic group $p$-adic group Jordan block endoparameter types and covers

Citation

Blondel, Corinne; Henniart, Guy; Stevens, Shaun. Jordan blocks of cuspidal representations of symplectic groups. Algebra Number Theory 12 (2018), no. 10, 2327--2386. doi:10.2140/ant.2018.12.2327. https://projecteuclid.org/euclid.ant/1550113225


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References

  • J. D. Adler, “Self-contragredient supercuspidal representations of ${\rm GL}_n$”, Proc. Amer. Math. Soc. 125:8 (1997), 2471–2479.
  • J. Arthur, The endoscopic classification of representations, Amer. Math. Soc. Colloq. Publ. 61, Amer. Math. Soc., Providence, RI, 2013.
  • C. Blondel, “${\rm Sp}(2N)$-covers for self-contragredient supercuspidal representations of ${\rm GL}(N)$”, Ann. Sci. École Norm. Sup. $(4)$ 37:4 (2004), 533–558.
  • C. Blondel, “Représentation de Weil et $\beta$-extensions”, Ann. Inst. Fourier $($Grenoble$)$ 62:4 (2012), 1319–1366.
  • C. Blondel and S. Stevens, “Genericity of supercuspidal representations of $p$-adic ${\rm Sp}_4$”, Compos. Math. 145:1 (2009), 213–246.
  • C. J. Bushnell and G. Henniart, “Local tame lifting for ${\rm GL}(N)$, I: Simple characters”, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 105–233.
  • C. J. Bushnell and G. Henniart, “Local tame lifting for ${\rm GL}(n)$, IV: Simple characters and base change”, Proc. London Math. Soc. $(3)$ 87:2 (2003), 337–362.
  • C. J. Bushnell and G. Henniart, “The essentially tame local Langlands correspondence, I”, J. Amer. Math. Soc. 18:3 (2005), 685–710.
  • C. J. Bushnell and G. Henniart, The local Langlands conjecture for ${\rm GL}(2)$, Grundlehren der Math. Wissenschaften 335, Springer, 2006.
  • C. J. Bushnell and G. Henniart, To an effective local Langlands correspondence, Mem. Amer. Math. Soc. 1087, 2014.
  • C. J. Bushnell and P. C. Kutzko, The admissible dual of ${\rm GL}(N)$ via compact open subgroups, Ann. of Math. Studies 129, Princeton Univ. Press, 1993.
  • C. J. Bushnell and P. C. Kutzko, “Smooth representations of reductive $p$-adic groups: structure theory via types”, Proc. London Math. Soc. $(3)$ 77:3 (1998), 582–634.
  • C. J. Bushnell and P. C. Kutzko, “Semisimple types in ${\rm GL}_n$”, Compos. Math. 119:1 (1999), 53–97.
  • M. Cabanes and M. Enguehard, Representation theory of finite reductive groups, New Math. Monographs 1, Cambridge Univ. Press, 2004.
  • J.-F. Dat, “Finitude pour les représentations lisses de groupes $p$-adiques”, J. Inst. Math. Jussieu 8:2 (2009), 261–333.
  • I. M. Gel'fand and D. A. Kajdan, “Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field”, pp. 95–118 in Lie groups and their representations (Budapest, 1971), edited by I. M. Gel'fand, Halsted, New York, 1975.
  • D. Goldberg, P. Kutzko, and S. Stevens, “Covers for self-dual supercuspidal representations of the Siegel Levi subgroup of classical $p$-adic groups”, Int. Math. Res. Not. 2007:22 (2007), art. id. rnm085.
  • B. H. Gross and M. Reeder, “Arithmetic invariants of discrete Langlands parameters”, Duke Math. J. 154:3 (2010), 431–508.
  • M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. of Math. Studies 151, Princeton Univ. Press, 2001.
  • G. Henniart, “Une preuve simple des conjectures de Langlands pour ${\rm GL}(n)$ sur un corps $p$-adique”, Invent. Math. 139:2 (2000), 439–455.
  • G. Henniart, “Correspondance de Langlands et fonctions $L$ des carrés extérieur et symétrique”, Int. Math. Res. Not. 2010:4 (2010), 633–673.
  • K. Kariyama, “On types for unramified $p$-adic unitary groups”, Canad. J. Math. 60:5 (2008), 1067–1107.
  • R. Kurinczuk, D. Skodlerack, and S. Stevens, “Endo-classes for $p$-adic classical groups”, preprint, 2016.
  • G. Laumon, M. Rapoport, and U. Stuhler, “${\mathscr D}$-elliptic sheaves and the Langlands correspondence”, Invent. Math. 113:2 (1993), 217–338.
  • J. Lust and S. Stevens, “On depth zero $L$-packets for classical groups”, preprint, 2016.
  • G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Studies 107, Princeton Univ. Press, 1984.
  • M. Miyauchi and S. Stevens, “Semisimple types for $p$-adic classical groups”, Math. Ann. 358:1-2 (2014), 257–288.
  • C. Mœglin, “Points de réductibilité pour les induites de cuspidales”, J. Algebra 268:1 (2003), 81–117.
  • C. Mœglin, “Multiplicité $1$ dans les paquets d'Arthur aux places $p$-adiques”, pp. 333–374 in On certain $L$-functions (West Lafayette, IN, 2007), edited by J. Arthur et al., Clay Math. Proc. 13, Amer. Math. Soc., Providence, RI, 2011.
  • C. Mœglin, “Paquets stables des séries discrètes accessibles par endoscopie tordue: leur paramètre de Langlands”, pp. 295–336 in Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro (New Haven, CT, 2012), edited by J. W. Cogdell et al., Contemp. Math. 614, Amer. Math. Soc., Providence, RI, 2014.
  • C. Mœglin and M. Tadić, “Construction of discrete series for classical $p$-adic groups”, J. Amer. Math. Soc. 15:3 (2002), 715–786.
  • V. Sécherre and S. Stevens, “Towards an explicit local Jacquet–Langlands correspondence beyond the cuspidal case”, preprint, 2016.
  • S. Stevens, “Intertwining and supercuspidal types for $p$-adic classical groups”, Proc. London Math. Soc. $(3)$ 83:1 (2001), 120–140.
  • S. Stevens, “Semisimple characters for $p$-adic classical groups”, Duke Math. J. 127:1 (2005), 123–173.
  • S. Stevens, “The supercuspidal representations of $p$-adic classical groups”, Invent. Math. 172:2 (2008), 289–352.