Skip to main content
SpringerLink
Log in

Arthur’s Classification for the Classical \(\mathbb {Z}\)-groups

  • Chapter
  • First Online:
Automorphic Forms and Even Unimodular Lattices

  • 1130 Accesses

Abstract

In this chapter, we explain Arthur’s description of the discrete automorphic representations of classical groups in terms of selfdual cuspidal automorphic representations of GLn. In agreement with the general philosophy of this book, we restrict our exposition to the level 1 automorphic representations, but provide a concrete form of the famous Arthur multiplicity formula for those representations whose Archimedean component is a discrete series. As a prerequisite, we discuss Shelstad’s parametrization of the individual elements of a discrete series L-packet, including several examples, and its generalization to the Adams-Johnson packets. We apply this theory to Siegel modular forms and orthogonal automorphic forms; this sheds much light on the more elementary constructions of the previous chapters. We also discuss applications to standard L-functions and Galois representations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In this chapter, we use the term Chevalley group as a synonym for split semisimple \(\mathbb {Z}\)-group.

  2. 2.

    The choice of this root is fixed once and for all and will not play any role in the rest of this book. For example, since all these roots are permuted transitively by W(D n+1) ⊂O(En+1), the isomorphism class of the \(\mathbb {Z}\)-group SOn depends only on n.

  3. 3.

    Let us mention that none of the difficulties stated by Arthur in his Chap. 9 seem to apply to the situation we are interested in, which concerns only “pure inner forms” of Chevalley groups [115].

  4. 4.

    The reader should be aware that there are several notions of algebraic automorphic representations in the literature. Definition 8.2.7, which is essentially the one considered, for example, in [33, Sect. 18.2], but which is not the one used by Clozel in [59], is reminiscent of the notion of Hecke character of type A0 in the sense of Weil (see [43] for a clarification of the various notions).

  5. 5.

    Following Langlands, it is suggestive to write \(z^\lambda \overline {z}^\mu \) for the element (z∕|z|)λ−μ|z|λ+μ when \(z \in \mathbb {C}^\times \) and \(\lambda ,\mu \in \mathbb {C}\) are such that \(\lambda - \mu \in \mathbb {Z}\).

  6. 6.

    In this definition of Pp(π), it is understood that cp(π) p w(π)∕2 denotes the semisimple conjugacy class of \({\mathrm{GL}}_n(\mathbb {C})\) obtained by taking the product of the class cp(π), viewed in \({\mathrm{GL}}_n(\mathbb {C}) \supset {\mathrm{SL}}_n(\mathbb {C})\), and the scalar \(p^{w(\pi )/2} \in \mathbb {C}^\ast \subset \mathrm {GL}_n(\mathbb {C})\).

  7. 7.

    What we denote here by \(\psi _{\mathbb {R}}\), \({\mathrm{C}}_{\nu _\infty }\), and \(\varPi (\psi _{\mathbb {R}})\) is denoted by ψ, \(\mathcal {S}_\psi \), and \(\widetilde {\varPi }_{\psi }\), respectively, in Arthur’s statement; moreover, Arthur does not give a name to ι and for u ∈ Π(ν ∞), he writes the character χ u as x↦〈x, u〉. The image of ι, a finite subset of \(\varPi _{\mathrm{unit}}(G(\mathbb {R}))\), is commonly called the Arthur packet associated with \(\psi _{\mathbb {R}}\).

  8. 8.

    When \(L_{\varDelta ,\varphi }(\mathbb {R})\) is connected, the character ρ is determined by its differential at the identity, itself characterized by the property that the representation π Δ,φ defined in the text must have the same infinitesimal character as V .

References

  1. J. Adams, Discrete series and characters of the component group, in Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, ed. by L. Clozel, M. Harris, J.-P. Labesse, and B.-C. Ngô (International Press, 2011), Chap. 4.2.

    Google Scholar 

  2. J. Adams, D. Barbasch, D. Vogan, The Langlands classification and irreducible characters for real reductive groups, Progr. Math., vol. 104 (Birkhaüser, Boston-Basel-Berlin, 1992).

    Google Scholar 

  3. J. Adams, J. Johnson, Endoscopic groups and packets of non-tempered representations, Compos. Math. 64 (1987), pp. 271–309.

    MATH  Google Scholar 

  4. A. N. Andrianov, V. L. Kalinin, On the analytic properties of standard zeta functions of Siegel modular forms, Math. USSR, Sb. 35 (1979), pp. 1–17.

    Article  Google Scholar 

  5. N. Arancibia, Paquets d’Arthur des représentations cohomologiques, Ph. D. thesis, Univ. Paris 6 (2015).

    Google Scholar 

  6. J. Arthur, Unipotent automorphic representations: conjectures, in Orbites unipotentes et représentations II : groupes p-adiques et réels, Astérisque 171–172 (Soc. Math. France, Paris, 1989), pp. 13–71.

    Google Scholar 

  7. J. Arthur, On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99, pp. 209–283 (1999).

    Article  MathSciNet  Google Scholar 

  8. J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publ., vol. 61 (Amer. Math. Soc., 2013).

    Google Scholar 

  9. J. Bellaïche, G. Chenevier, The sign of Galois representations attached to automorphic forms for unitary groups, Compos. Math. 147 (2011), pp. 1337–1352.

    Article  MathSciNet  Google Scholar 

  10. S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. reine angew. Math. 362 (1985), pp. 146–168.

    MathSciNet  MATH  Google Scholar 

  11. A. Borel, Automorphic L-functions, in Automorphic forms, representations and L-functions, II (Oregon State Univ., Corvallis, Ore.), Proc. Symp. in Pure Math. XXXIII (Amer. Math. Soc., Providence, RI, 1979), pp. 27–61.

    Google Scholar 

  12. M. Borovoi, Galois cohomology of reductive algebraic groups over the field of real numbers, preprint available at http://arxiv.org/abs/1401.5913.

  13. K. Buzzard & T. Gee, The conjectural connections between automorphic representations and Galois representations, Automorphic forms and Galois representations Vol. 1, London Math. Soc. Lecture Note Ser. 414, Cambridge Univ. Press, Cambridge (2014), pp 135–187.

    Google Scholar 

  14. A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J. 161 (2010), pp. 2311–2413.

    Article  MathSciNet  Google Scholar 

  15. P.-H. Chaudouard, G. Laumon, Le lemme fondamental pondéré. I. Constructions géométriques, Compos. Math. 146 (2010), pp. 1416–1506.

    Article  MathSciNet  Google Scholar 

  16. P.-H. Chaudouard, G. Laumon, Le lemme fondamental pondéré. II. Énoncés cohomologiques, Annals of Math. 176 (2012), pp. 1647–1781.

    Article  MathSciNet  Google Scholar 

  17. G. Chenevier, M. Harris, Construction of automorphic Galois representations II, Cambridge Math. J. 1 (2013), pp. 53–73.

    Article  MathSciNet  Google Scholar 

  18. G. Chenevier, D. Renard, Level one algebraic cusp form of classical groups of small rank, Mem. Amer. Math. Soc., vol. 1121 (Amer. Math. Soc., Providence, RI, 2015).

    Google Scholar 

  19. L. Clozel, Motifs et formes automorphes : applications du principe de fonctorialité, in Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI., 1988), Perspectives in Math. vol. 10 (Academic Press, Boston, MA, 1990).

    Google Scholar 

  20. L. Clozel, Purity Reigns Supreme, Int. Math. Res. Not., I.M.R.N., vol. 2013 (2013), pp. 328–346.

    Article  MathSciNet  Google Scholar 

  21. L. Clozel, M. Harris, J.-P. Labesse, Construction of automorphic Galois representations I, in Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, ed. by L. Clozel, M. Harris, J.-P. Labesse, B.-C. Ngô (International Press, 2011).

    Google Scholar 

  22. J. Cogdell, Lectures on L-functions, converse theorems, and functoriality for GL n, in Lectures on automorphic L-functions, Fields Inst. Monogr. (Amer. Math. Soc., Providence, RI, 2004).

    Google Scholar 

  23. D. Ginzburg, S. Rallis, D. Soudry, The descent map from automorphic representations of GL(n) to classical groups (World Sci. Publishing, 2011).

    Google Scholar 

  24. W. Goldring, Galois representations associated to holomorphic limits of discrete series I: unitary groups, Compos. Math. 150 (2014), pp. 191–228.

    Article  MathSciNet  Google Scholar 

  25. T. Ikeda, Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture, Duke Math. J. 131 (2006), pp. 469–497.

    Article  MathSciNet  Google Scholar 

  26. T. Ikeda, On the lifting of automorphic representations from \(\mbox{PGL}_2(\mathbb {A})\) to \(\mbox{Sp}_{2n}(\mathbb {A})\) or \(\widetilde {\mbox{Sp}_{2n+1}(\mathbb {A})}\) over a totally real field, available at https://www.math.kyoto-u.ac.jp/~ikeda/.

  27. H. Jacquet, J. Shalika, A non-vanishing theorem for zeta-functions of GL n, Invent. Math. 38 (1976), pp. 1–16.

    Article  MathSciNet  Google Scholar 

  28. T. Kaletha, Rigid inner forms of real and p-adic groups, preprint available at http://arxiv.org/abs/1304.3292.

  29. A. W. Knapp, Representation theory of semisimple groups (Princeton Univ. Press, 1986).

    Google Scholar 

  30. A. Knapp, Local Langlands correspondence: the archimedean case, in Motives vol. II, Proc. Sympos. Pure Math., vol. 55 (Amer. Math. Soc., Providence, RI, 1994), pp. 393–410.

    Google Scholar 

  31. A. Knapp & G. Zuckerman, Normalizing factors, tempered representations and L-groups, in Automorphic forms, representations and L-functions, I (Oregon State Univ., Corvallis, Ore.), Proc. Symp. in Pure Math. XXXIII (Amer. Math. Soc., Providence, RI, 1979), pp. 93–105.

    Google Scholar 

  32. R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), pp. 611–650.

    Article  MathSciNet  Google Scholar 

  33. R. Kottwitz, Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties, and L-functions vol. I (Ann Arbor, MI, 1998), Perspect. Math., vol. 10 (Academic Press, 1990), pp. 161–209.

    Google Scholar 

  34. J.-P. Labesse, R. Langlands, L-indistinguishability for SL(2), Canadian J. Math. 31 (1979), pp. 726–785.

    Article  MathSciNet  Google Scholar 

  35. J.-P. Labesse, J.-L. Waldspurger, La formule des traces tordue d’après le Friday Morning Seminar, CRM Monogr. Ser., vol 31 (Amer. Math. Soc., Providence, RI, 2013).

    Google Scholar 

  36. R. Langlands, The classification of representations of real reductive groups, Math. Surveys Monogr., vol. 31 (Amer. Math. Soc., Provindence, RI, 1973).

    Google Scholar 

  37. R. Langlands, On the functional equation satisfied by Eisenstein series, Lecture Notes in Math., vol. 544 (Springer Verlag, 1976).

    Google Scholar 

  38. R. Langlands, Automorphic representations, Shimura varieties, and motives (Ein Märchen), in Automorphic forms, representations and L-functions, II (Oregon State Univ., Corvallis, Ore.), Proc. Symp. in Pure Math. XXXIII (Amer. Math. Soc., Providence, RI, 1979), pp. 205–246.

    Google Scholar 

  39. R. Langlands, D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), pp. 219–271.

    Article  MathSciNet  Google Scholar 

  40. P. Mezo, Character identities in the twisted endoscopy of real reductive groups, Mem. Amer. Math. Soc., vol. 222 (Amer. Math. Soc. Provindence, RI, 2013).

    Google Scholar 

  41. P. Mezo, Tempered spectral transfer in the twisted endoscopy of real groups, preprint available at http://people.math.carleton.ca/~mezo/ (2013).

  42. S.-I. Mizumoto, Poles and residues of standard L-functions attached to Siegel modular forms, Math. Ann. 289 (1991), pp. 589–612.

    Article  MathSciNet  Google Scholar 

  43. C. Moeglin, J.-L. Waldspurger, Stabilisation de la formule des traces tordue VI & X, preprint available at http://webusers.imj-prg.fr/~jean-loup.waldspurger/ (2014).

  44. Motives, ed. by U. Jannsen, S. Kleiman, J.-P. Serre, Proc. Symp. in Pure Math., vol. 55 (Amer. Math. Soc., 1991).

    Google Scholar 

  45. B. C. Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2009), pp. 1–169.

    Article  Google Scholar 

  46. I. Piatetski-Shapiro, S. Rallis, L-functions for the classical groups, Lecture Notes in Math., vol. 1254 (Springer Verlag, 1987), pp. 1–52.

    Google Scholar 

  47. J.-P. Serre, Cohomologie galoisienne, 5th edn., Lecture Notes in Math., vol. 5 (Springer Verlag, 1994).

    Google Scholar 

  48. D. Shelstad, Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. Éc. Norm. Sup. 12 (1979), pp. 1–31.

    Article  MathSciNet  Google Scholar 

  49. D. Shelstad, L-indistinguishability for real groups, Math. Ann. 259 (1982), pp. 385–430.

    Article  MathSciNet  Google Scholar 

  50. D. Shelstad, Tempered endoscopy for real groups III : inversion of transfer and L-packet structure, Represent. Theory 12 (2008), pp. 369–402.

    Article  MathSciNet  Google Scholar 

  51. D. Shelstad, Examples in endoscopy for real groups, notes from a talk at the conference Stable trace formula (Banff, 2008), available at http://andromeda.rutgers.edu/~shelstad/.

  52. D. Shelstad, Some results on endoscopic transfer, notes from a talk at the conference L-packets (Banff, 2011), available at http://andromeda.rutgers.edu/~shelstad/.

  53. D. Shelstad, On geometric transfer in real twisted endoscopy, Ann. of Math. 176 (2012), pp. 1919–1985.

    Article  MathSciNet  Google Scholar 

  54. D. Shelstad, On the structure of endoscopic transfer factors, preprint available at http://andromeda.rutgers.edu/~shelstad/.

  55. D. Shelstad, On spectral transfer factors in real twisted endoscopy, preprint available at http://andromeda.rutgers.edu/~shelstad/.

  56. S.-W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. 173 (2011), pp. 1645–1741.

    Article  MathSciNet  Google Scholar 

  57. O. Taïbi, Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, preprint available at http://arxiv.org/abs/1406.4247 (2014).

  58. J. Tate, Number theoretic background, in Automorphic forms, representations and L-functions, II (Oregon State Univ., Corvallis, Ore.), Proc. Symp. in Pure Math. XXXIII (Amer. Math. Soc., Providence, RI, 1979), pp. 3–26.

    Google Scholar 

  59. D. Vogan, Representations of real reductive Lie groups, Progr. Math., vol. 15 (1981).

    Google Scholar 

  60. D. Vogan, On the unitarizability of certain series of representations, Annals of Math. 120 (1984), pp. 141–187.

    Article  MathSciNet  Google Scholar 

  61. D. Vogan, G. Zuckerman, Unitary representations with non-zero cohomology, Compos. Math. 53 (1984), pp. 51–90.

    MATH  Google Scholar 

  62. J.-L. Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), pp. 423–525.

    Article  MathSciNet  Google Scholar 

  63. J.-L. Waldspurger, Stabilisation de la formule des traces tordues I, II, III, IV, V, VII, VIII, IX, preprints available at http://webusers.imj-prg.fr/~jean-loup.waldspurger/ (2014).

Download references

Author information

Authors and Affiliations

  1. CNRS, Institut de Mathématique d’Orsay, Université Paris-Sud, Orsay, France

    Gaëtan Chenevier

  2. Institut de Mathématiques de Jussieu, Université Paris Diderot, Paris, France

    Jean Lannes

Authors
  1. Gaëtan Chenevier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean Lannes
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chenevier, G., Lannes, J. (2019). Arthur’s Classification for the Classical \(\mathbb {Z}\)-groups. In: Automorphic Forms and Even Unimodular Lattices. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-95891-0_8

Download citation

Publish with us

Policies and ethics

Search

Navigation