Advertisement

Local Langlands and Springer Correspondences

  • Anne-Marie AubertEmail author
Chapter
Part of the Progress in Mathematics book series (PM, volume 328)

Abstract

These notes give an overview of results obtained jointly with Ahmed Moussaoui and Maarten Solleveld on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld.

2010 Mathematics Subject Classification

20C08 14F43 20G20 

References

  1. 1.
    P. Achar, A. Henderson, D. Juteau, S. Riche, Modular generalized Springer correspondence I: the general linear group. J. Eur. Math. Soc. (JEMS) 18(7), 1405–1436 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Arthur, A note on \(L\)-packets. Pure Appl. Math. Q. 2(1), 199–217 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Colloquium Publications, vol. 61 (American Mathematical Society, 2013)Google Scholar
  4. 4.
    M. Asgari, K. Choiy, The local Langlands conjecture for \(p\)-adic \({\rm GSpin}_4\), \({\rm GSpin}_6\), and their inner forms. Forum Math. 29(6), 1261–1290 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.-M. Aubert, Around the Langlands program. Jahresber. Dtsch. Math.-Ver. 120(1), 3–40 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, On the local Langlands correspondence for non-tempered representations. Münster J. Math. 7(1), 27–50 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Geometric structure in smooth dual and local Langlands correspondence. Japan. J. Math. 9, 99–136 (2014)CrossRefGoogle Scholar
  8. 8.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Depth and the local Langlands correspondence, in Arbeitstagung Bonn 2013, in Memory of Friedrich Hirzebruch W. Ballmann, C. Blohmann, G. Faltings, P. Teichner, et D. Zagier, Progr. Math. vol. 319 (Birkhäuser/Springer, Berlin, 2016), pp. 17–41Google Scholar
  9. 9.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The local Langlands correspondence for inner forms of \({\rm SL}_n\). Res. Math. Sci. 3 (2016), Paper No. 32, 34 pGoogle Scholar
  10. 10.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Hecke algebras for inner forms of \(p\)-adic special linear groups. J. Inst. Math. Jussieu 16(2), 351–419 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The principal series of \(p\)-adic groups with disconnected centre. Proc. Lond. Math. Soc. (3) 114(5), 798–854 (2017)Google Scholar
  12. 12.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, Conjectures about \(p\)-adic groups and their noncommutative geometry, in Around Langlands Correspondences, Contemp. Math. vol. 691 (Amer. Math. Soc., Providence, RI, 2017), pp. 15–51Google Scholar
  13. 13.
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The noncommutative geometry of inner forms of \(p\)-adic special linear groups. arXiv:1505.04361 [math.RT]
  14. 14.
    A.-M. Aubert, S. Mendes, R. Plymen, M. Solleveld, \(L\)-packets and depth for \({\rm SL}_2(K)\) with \(K\) a local function field of characteristic \(2\). Int. J. Number Theory 13(10), 2545–2568 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.-M. Aubert, A. Moussaoui, M. Solleveld, Generalizations of the Springer correspondence and cuspidal Langlands parameters. Manuscripta Math. 1–72 (2018).  https://doi.org/10.1007/s00229-018-1001-8MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.-M. Aubert, A. Moussaoui, M. Solleveld, Graded Hecke algebras for disconnected reductive groups, to appear in the Proceedings of the Simons Symposium 2017, Geometric Aspects of the Trace FormulaGoogle Scholar
  17. 17.
    A.-M. Aubert, A. Moussaoui, M. Solleveld, Affine Hecke algebras for Langlands parameters. arXiv:1701.03593
  18. 18.
    J. Bernstein, P. Deligne, Le “centre” de Bernstein, in Représentations des groupes réductifs sur un corps local (Travaux en cours, Hermann, 1984), pp. 1–32Google Scholar
  19. 19.
    J. Bernstein, V. Lunts, Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578 (Springer, Berlin, 1994)CrossRefGoogle Scholar
  20. 20.
    J. Bernstein, A. Zelevinsky, representations of the group \({\rm GL}(n, F)\) where \(F\) is a non-Archimedean local field. Russian Math. Surveys 31(3), 1–68 (1976)CrossRefGoogle Scholar
  21. 21.
    A. Borel, Automorphic \(L\)-functions. Proc. Symp. Pure Math 33(2), 27–61 (1979)MathSciNetCrossRefGoogle Scholar
  22. 22.
    C. Bushnell, G. Henniart, The essentially tame local Langlands correspondence, III: the general case. Proc. Lond. Math. Soc. (3) 101(2), 497–553 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    K. Choiy, The local Langlands conjecture for the \(p\)-adic inner form of \({\rm Sp}_4\). Int. Math. Res. Not. IMRN 6, 1830–1889 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, vol 11 (Wiley, New York, 1962)Google Scholar
  25. 25.
    S. DeBacker, M. Reeder, Depth-zero supercuspidal \(L\)-packets and their stability. Ann. of Math. (2) 169(3), 795–901 (2009)Google Scholar
  26. 26.
    Y. Feng, E. Opdam, M. Solleveld, Supercuspidal unipotent representations: \(L\)-packets and formal degrees. arXiv:1805.01888
  27. 27.
    W.T. Gan, S. Takeda, The local Langlands conjecture for \({\rm GSp}(4)\). Ann. Math. (2) 173(3), 1841–1882 (2011)Google Scholar
  28. 28.
    W.T. Gan, S. Takeda, The local Langlands conjecture for \({\rm Sp}(4)\). Int. Math. Res. Not. IMRN 15, 2987–3038 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    W.T. Gan, W. Tantono, The local Langlands conjecture for \({\rm GSp}(4)\), II: the case of inner forms. Am. J. Math. 136(3), 761–805 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    R. Ganapathy, The local Langlands correspondence for \({\rm GSp}_4\) over local function fields. Amer. J. Math. 137(6), 1441–1534 (2015)Google Scholar
  31. 31.
    R. Ganapathy, S. Varma, On the local Langlands correspondence for split classical groups over local function fields. J. Inst. Math. Jussieu 16(5), 987–1074 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    R. Godement, H. Jacquet, Zeta Functions of Simple Algebras. Lecture Notes in Mathematics, vol. 260 (Springer, Berlin, 1972)Google Scholar
  33. 33.
    T.J. Haines, The stable Bernstein center and test functions for Shimura varieties, in Automorphic Forms and Galois Representations, London Mathematical Society Lecture Note Series, vol. 415 (Cambridge University Press, Cambridge, 2014), pp. 118–186Google Scholar
  34. 34.
    M. Harris, R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Math. Studies, vol. 151 (Princeton University Press, Princeton, 2001)Google Scholar
  35. 35.
    G. Henniart, Une preuve simple des conjectures de Langlands pour \({\rm GL}(n)\) sur un corps \(p\)-adique. Inventiones Mathematicae 139, 439–455 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    K. Hiraga, H. Saito, On \(L\)-packets for inner forms of \({\rm SL}_n\). Mem. Amer. Math. Soc. 1013, vol. 215 (2012)Google Scholar
  37. 37.
    D. Jiang, C. Nien, On the local Langlands conjecture and related problems over \(p\)-adic local fields, in Proceedings of the Sixth International Congress of Chinese Mathematicians vol I, 309–325, Adv. Lect. Math. (ALM), vol. 36 (Int. Press, Somerville, MA, 2017)Google Scholar
  38. 38.
    D. Jiang, C. Nien, The local Langlands conjectures for non-quasi-split groups, in Families of Automorphic Forms and the Trace Formula (Springer, Berlin, 2016), pp. 217–257 (Simons Symp.)Google Scholar
  39. 39.
    T. Kaletha, Global rigid inner forms and multiplicities of discrete automorphic representations. Invent. Math. 213(1), 271–369 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    T. Kaletha, A. Minguez, S.W. Shin, P.-J. White, Endoscopic classification of representations: inner forms of unitary groups. arXiv:1409.3731
  41. 41.
    R. Kottwitz, Stable trace formula: cuspidal tempered terms. Duke Math. J. 51(3), 611–650 (1984)MathSciNetCrossRefGoogle Scholar
  42. 42.
    G. Laumon, M. Rapoport, U. Stuhler, \(\cal{D}\)-elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–238 (1993)MathSciNetCrossRefGoogle Scholar
  43. 43.
    L. Lomeli, Langlands program and Ramanujan conjecture: a survey (Preprint, 2018)Google Scholar
  44. 44.
    G. Lusztig, Some examples of square-integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277, 623–653 (1983)MathSciNetzbMATHGoogle Scholar
  45. 45.
    G. Lusztig, Intersection cohomology complexes on a reductive group. Invent. Math. 75(2), 205–272 (1984)MathSciNetCrossRefGoogle Scholar
  46. 46.
    G. Lusztig, Fourier transforms on a semisimple Lie algebra over \({\mathbb{F}_q}\), in Algebraic GroupsGoogle Scholar
  47. 47.
    G. Lusztig, Classification of unipotent representations of simple \(p\)-adic groups. Int. Math. Res. Notices 11, 517–589 (1995)MathSciNetCrossRefGoogle Scholar
  48. 48.
    G. Lusztig, Classification of unipotent representations of simple \(p\)-adic groups. II. Represent. Theory 6, 243–289 (2002)MathSciNetCrossRefGoogle Scholar
  49. 49.
    G. Lusztig, Character sheaves on disconnected groups. V. Represent. Theory 8, 346–376 (2004)MathSciNetCrossRefGoogle Scholar
  50. 50.
    M. Mishra, B. Pattanayak, A note on depth preservation. To appear in J Ramanujan Math. Soc. arXiv:1903.10771
  51. 51.
    C.P. Mok, Endoscopic classification of representations of quasi-split unitary groups. Mem. Amer. Math. Soc. 235 (2015), no. 1108, vi+248 pMathSciNetCrossRefGoogle Scholar
  52. 52.
    A. Moussaoui, Centre de Bernstein dual pour les groupes classiques. Represent. Theory 21, 172–246 (2017)MathSciNetCrossRefGoogle Scholar
  53. 53.
    A. Moussaoui, Proof of the Aubert-Baum-Plymen-Solleveld conjecture for split classical groups, in Around Langlands correspondences, Contemp. Math., vol. 691 (Amer. Math. Soc., Providence, RI, 2017), pp. 257–281Google Scholar
  54. 54.
    A. Moy, G. Prasad, Jacquet functors and unrefined minimal \(K\)-types. Comment. Math. Helv. 71(1), 98–121 (1996)MathSciNetCrossRefGoogle Scholar
  55. 55.
    M. Oi, Depth preserving property of the local Langlands correspondence for quasi-split classical groups in a large residual characteristic. arXiv:1804.10901
  56. 56.
    P. Scholze, The local Langlands correspondence for \({\rm GL}_n\) over \(p\)-adic fields. Invent. Math. 192, 663–715 (2013)MathSciNetCrossRefGoogle Scholar
  57. 57.
    J.-P. Serre, Corps Locaux (Hermann, Paris, 1962)Google Scholar
  58. 58.
    A. Silberger, W. Zink, Langlands classification for \(L\)-parameters. J. AlgebraGoogle Scholar
  59. 59.
    T.A. Springer, Linear Algebraic Groups 2nd ed., Progress in Mathematics, vol. 9 (Birkhäuser, 1998)Google Scholar
  60. 60.
    J. Tate, Number theoretic background. Proc. Symp. Pure Math 33(2), 3–26 (1979)MathSciNetCrossRefGoogle Scholar
  61. 61.
    D. Vogan, The local Langlands conjecture, in Representation Theory of Groups and Algebras, Contemp. Math., vol. 145 (American Mathematical Society, 1993), pp. 305–379Google Scholar
  62. 62.
    B. Xu, On a lifting problem of \(L\)-packets. Compos. Math. 152(9), 1800–1850 (2016)MathSciNetCrossRefGoogle Scholar
  63. 63.
    J.-K. Yu, On the local Langlands correspondence for tori, in Ottawa Lectures on Admissible Representations of Reductice p-adic Groups, Fields Institute Monographs (American Mathemical Society, 2009), pp. 177–183Google Scholar
  64. 64.
    A.V. Zelevinsky, Induced representations of reductive \(p\)-adic groups II. On irreducible representations of \({\rm GL}(n)\), Ann. Sci. École Norm. Sup. (4) 13(2), 165–210 (1980)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu—Paris Rive Gauche, IMJ-PRGC.N.R.S, Sorbonne Université, Université Paris DiderotParisFrance

Personalised recommendations