Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of G 2

Jian-Shu Li; Joachim Schwermer

Compositio Mathematica (1993)

  • Volume: 87, Issue: 1, page 45-78
  • ISSN: 0010-437X

How to cite


Li, Jian-Shu, and Schwermer, Joachim. "Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of $G_2$." Compositio Mathematica 87.1 (1993): 45-78. <>.

author = {Li, Jian-Shu, Schwermer, Joachim},
journal = {Compositio Mathematica},
keywords = {automorphic forms; non-vanishing results; cohomology classes; arithmetic subgroup; global theta lifting; Eisenstein series; irreducible automorphic representation; relative Lie algebra cohomology},
language = {eng},
number = {1},
pages = {45-78},
publisher = {Kluwer Academic Publishers},
title = {Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of $G_2$},
url = {},
volume = {87},
year = {1993},

AU - Li, Jian-Shu
AU - Schwermer, Joachim
TI - Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of $G_2$
JO - Compositio Mathematica
PY - 1993
PB - Kluwer Academic Publishers
VL - 87
IS - 1
SP - 45
EP - 78
LA - eng
KW - automorphic forms; non-vanishing results; cohomology classes; arithmetic subgroup; global theta lifting; Eisenstein series; irreducible automorphic representation; relative Lie algebra cohomology
UR -
ER -


  1. [1] Adams, J.: Discrete spectrum of the reductive dual pair (0(p, q), Sp(2m)). Invent. Math.74 (1983), 449-475. Zbl0561.22011MR724015
  2. [2] Borel, A.: Stable real cohomology of arithmetic groups II. In J. Hano et al. eds. Manifolds and Lie Groups. Progress in Maths. vol. 14, pp. 21-55, Boston1981. Zbl0483.57026MR642850
  3. [3] Borel, A.: Automorphic L-functions. In Proc. Symp. Pure Maths., vol. 33, II, pp. 27-61, Providence1979. Zbl0412.10017MR546608
  4. [4] Borel, A., Casselman, W.: L 2-cohomology of locally symmetric manifolds of finite volume. Duke Math. J.50 (1983), 625-647. Zbl0528.22012MR714821
  5. [5] Borel, A., Garland, H.: Laplacian and the discrete spectrum of an arithmetic group. American J. of Maths.105 (1983), 309-335. Zbl0572.22007MR701563
  6. [6] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Studies94, Princeton1980. Zbl0443.22010MR1721403
  7. [7] Deligne, P.: La conjecture de Weil I, Publ. IHES43 (1973), 273-307. Zbl0287.14001MR340258
  8. [8] Gelbart, S.: Weil's Representation and the Spectrum of the Metaplectic Group. Lecture Notes in Maths. Vol. 530. Berlin-Heidelberg-New York1976. Zbl0365.22017
  9. [9] Godement, R., Jacquet, H.: Zeta Functions of Simple Algebras. Lecture Notes in Maths., Vol. 260. Berlin -Heidelberg-New York1972. Zbl0244.12011MR342495
  10. [10] Harder, G.: On the cohomology of discrete arithmetically defined groups. In Proc. of the Int. Colloq. on Discrete Subgroups of Lie groups and Appl. to Moduli (Bombay1973), pp. 129-160, Oxford1975. Zbl0317.57022MR425018
  11. [11] Harder, G.: On the cohomology of SL2(O). In Lie groups and their representations, Proc. of the summer school on group representations, pp. 139-150, London1975. Zbl0395.57028MR425019
  12. [12] Harder, G.: Eisenstein cohomology of arithmetic groups: The case GL2. Inventiones Math.89 (1987), 37-118. Zbl0629.10023MR892187
  13. [13] Howe, R.: θ-series and invariant theory. In Automorphic Forms, Representations and L-functions, Proc. Symp. Pure Maths, Vol. 33, I, pp. 275-285, Providence1979. Zbl0423.22016
  14. [14] Howe, R.: On some results of Strichartz and of Rallis and Schiffmann. J. Functional Analysis32 (1979), 297-303. Zbl0408.22018MR538856
  15. [15] Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Berlin-Heidelberg- New York1972. Zbl0254.17004MR323842
  16. [16] Jacobson, N.: Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo (2) 7 (1958), 55-80. Zbl0083.02702MR101253
  17. [17] Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I, American J. of Maths.103 (1981), 499-558. Zbl0473.12008MR618323
  18. [18] Jacquet, H., Shalika, J.A.: A non-vanishing theorem for zeta functions of GLn. Invent. Math.38 (1976), 1-16. Zbl0349.12006MR432596
  19. [19] Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representation and harmonic polynomials. Invent. Math.44 (1978), 1-47. Zbl0375.22009MR463359
  20. [20] Kazhdan, D.: Some applications of the Weil representation. J. D'Analyse Math.32 (1977), 235-248. Zbl0445.22018MR492089
  21. [21] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math.74 (1961), 329-387. Zbl0134.03501MR142696
  22. [22] Labesse, J.-P., Schwermer, J.: On liftings and cusp cohomology of arithmetic groups. Invent. Math.83 (1986), 383-401. Zbl0581.10013MR818358
  23. [23] Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Maths. Vol. 544. Berlin-Heidelberg -New York1976. Zbl0332.10018MR579181
  24. [24] Langlands, R.P.: Euler products. New Haven1971. Zbl0231.20017MR419366
  25. [25] Li, J.-S.:Singular unitary representations of classical groups. Invent. Math.97 (1989), 237-255. Zbl0694.22011MR1001840
  26. [26] Li, J.-S.:Theta lifting for unitary representations with non-zero cohomology. Duke Math. J.61 (1990), 913-937. Zbl0723.22011MR1084465
  27. [27] Li, J.-S.: Non-vanishing theorems for the cohomology of certain arithmetic quotients. J. reine angew. Math.428 (1992), 177-217. Zbl0749.11032
  28. [28] Margulis, G.A.: Arithmeticity of irreducible lattices in semi simple groups of rank greater than 1. Invent. Math.76 (1984), 93-120. Zbl0551.20028MR739627
  29. [29] Moeglin, C.: Correspondence de Howe pour les paires reductives duales. Quelques calculs dans le cas archimédien. J. Functional Analysis85 (1989), 1-85. Zbl0729.22017MR1005856
  30. [30] Moeglin, C., Vigneras, M.F., Waldspurger, J.L.: Correspondances de Howe sur un Corps p-adique. Lecture Notes in Maths. Vol. 1291. Berlin-Heidelberg-New York1987. Zbl0642.22002MR1041060
  31. [31] Moeglin, C., Waldspurger, J.-L.: Décomposition Spectrale et Series d'Eisenstein-Une paraphrase de l'Ecriture -Paris1991. Zbl0794.11022
  32. [32] Oda, T., Schwermer, J.: Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two. Math. Ann.286 (1990), 481-509. Zbl0678.10022MR1032942
  33. [33] Rallis, S.: Injectivity properties of liftings associated to Weil representations. Compositio Math.52 (1984), 139-169. Zbl0624.22012MR750352
  34. [34] Rallis, S., Schiffmann, G.: Discrete spectrum of the Weil representation. Bull. AMS83 (1977), 267-270. Zbl0397.22009MR429753
  35. [35] Rallis, S., Schiffmann, G.: Automorphic cusp forms constructed from the Weil representation. Bull. AMS83 (1977), 271-275. Zbl0397.10022MR429754
  36. [36] Rallis, S., Schiffmann, G.: Weil Representation. I. Intertwining Distributions and Discrete Spectrum. Memoirs AMS, Vol. 231, Providence1980. Zbl0442.22006MR567800
  37. [37] Rallis, S., Schiffmann, G.: Representations supercuspidales du groupe metaplectique. J. Math. Kyoto Univ.17 (1977), 567-603. Zbl0398.22023MR498395
  38. [38] Rallis, S., Schiffmann, G.: Theta correspondence associated to G 2. American J. of Maths.111 (1989), 801-849. Zbl0723.11026MR1020830
  39. [39] Saito, M.: Representations unitaires des groupes symplectiques. J. Math. Soc. Japan24 (1972), 232-251. Zbl0232.22025MR299728
  40. [40] Satake, I.: Spherical functions and Ramanujan conjecture. In Proc. Symp. Pure Maths., Vol. 9, pp. 258-264, Providence1966. Zbl0202.41102MR211955
  41. [41] Schafer, R.D.: An Introduction to Non Associative Algebras, New York-London1966. Zbl0145.25601
  42. [42] Schwermer, J.: Kohomologie Arithmetisch Definierter Gruppen und Eisensteinreihen. Lecture Notes in Maths., Vol. 988, Berlin- Heidelberg-New York1983. Zbl0506.22015MR822473
  43. [43] Schwermer, J.: On arithmetic quotients of the Siegel upper half space of degree two. Compositio Math.58 (1986), 233-258. Zbl0596.10029MR844411
  44. [44] Schwermer, J.: Cohomology of arithmetic groups, automorphic forms and L-functions. In Cohomology of Arithmetic Groups and Automorphic Forms. Lecture Notes in Maths., Vol. 1447, pp. 1-29Berlin- Heidelberg-New York1990. Zbl0715.11028MR1082960
  45. [45] Schwermer, J.: On residues of Eisenstein series and associated cohomology classes for arithmetic groups. In preparation. Zbl0807.11031
  46. [46] Silberger, A.: Introduction to Harmonic Analysis on Reductive p-adic Groups, Mathematical Notes, Vol. 23, Princeton1979. Zbl0458.22006MR544991
  47. [47] Shahidi, F.: On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. Maths.127 (1988), 547-584. Zbl0654.10029MR942520
  48. [48] Shahidi, F.: Third symmetric power L-functions for GL(2). Compositio Math.70 (1989), 245-273. Zbl0684.10026MR1002045
  49. [49] Shahidi, F.: Langlands' conjecture on Plancherel measures for p-adic groups. In W. Barker and P. Sally eds., Harmonic Analysis on Reductive Groups, Progress in Maths., Vol. 101, pp. 277-295, Boston1991. Zbl0852.22017MR1168488
  50. [50] Shahidi, F.: On certain L-functions, Amer. J. Math.103 (1981), 297-356. Zbl0467.12013MR610479
  51. [51] Vogan, D.A. Jr.: Representations of Real Reductive Lie Groups. Boston-Basel-Stuttgart1981. Zbl0469.22012MR632407
  52. [52] Vogan, D.A. Jr., Zuckerman, G.J.: Unitary representations with non-zero cohomology. Compositio Math.53 (1984), 51-90. Zbl0692.22008MR762307
  53. [53] Weil, A.: Basic Number Theory, New York1974. Zbl0326.12001MR234930

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