Unitary representations with non-zero cohomology

David A. Vogan; Gregg J. Zuckerman

Compositio Mathematica (1984)

  • Volume: 53, Issue: 1, page 51-90
  • ISSN: 0010-437X

How to cite


Vogan, David A., and Zuckerman, Gregg J.. "Unitary representations with non-zero cohomology." Compositio Mathematica 53.1 (1984): 51-90. <http://eudml.org/doc/89677>.

author = {Vogan, David A., Zuckerman, Gregg J.},
journal = {Compositio Mathematica},
keywords = {automorphic forms; cohomology of locally symmetric spaces; infinite- dimensional representations; semisimple group; reductive Lie group; Lie algebra; unitary irreducible representations; Harish-Chandra modules; vanishing theorem for cohomology},
language = {eng},
number = {1},
pages = {51-90},
publisher = {Martinus Nijhoff Publishers},
title = {Unitary representations with non-zero cohomology},
url = {http://eudml.org/doc/89677},
volume = {53},
year = {1984},

AU - Vogan, David A.
AU - Zuckerman, Gregg J.
TI - Unitary representations with non-zero cohomology
JO - Compositio Mathematica
PY - 1984
PB - Martinus Nijhoff Publishers
VL - 53
IS - 1
SP - 51
EP - 90
LA - eng
KW - automorphic forms; cohomology of locally symmetric spaces; infinite- dimensional representations; semisimple group; reductive Lie group; Lie algebra; unitary irreducible representations; Harish-Chandra modules; vanishing theorem for cohomology
UR - http://eudml.org/doc/89677
ER -


  1. [1] M.W. Baldoni-Silva and D. Barbasch: The unitary spectrum for real rank one groups. Invent. Math.72 (1983) 27-55. Zbl0561.22009MR696689
  2. [2] A. Borel and N. Wallach: Continuous cohomology, discrete subgroups, representations of reductive groups, Princeton University Press, Princeton, New Jersey (1980). Zbl0443.22010MR554917
  3. [3] W. Casselman and M.S. Osborne: The n-cohomology of representations with an infinitesimal character. Comp. Math.31 (1975) 219-227. Zbl0343.17006MR396704
  4. [4] T. Enright: Relative Lie algebra cohomology and unitary representations of complex Lie groups. Duke Math. J.46 (1979) 513-525. Zbl0427.22010MR544243
  5. [5] A. Guichardet: Cohomologie des groupes topologiques et des algèbres de Lie, CEDIC-Fernand Nathan, Paris (1980). Zbl0464.22001MR644979
  6. [6] Harish-Chandra: Representations of semi-simple Lie groups I. Trans. Amer. Math. Soc.75 (1953) 185-243. Zbl0051.34002MR56610
  7. [7] S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978). Zbl0451.53038MR514561
  8. [8] R. Hotta and R. Parthasarathy: A geometric meaning of the multiplicities of integrable discrete classes in L2(Γ). Osaka J. Math.10 (1973) 211-234. Zbl0337.22016
  9. [9] J. Humphreys: Introduction to Lie algebras and representation theory. Springer-Verlag, New YorkHeidelbergBerlin (1972). Zbl0254.17004MR323842
  10. [10] S. Kumaresan: The canonical f-types of the irreducible unitary g-modules with non-zero relative cohomology. Invent. Math.59 (1980) 1-11. Zbl0442.22010MR575078
  11. [11] R. Parthasarathy: Dirac operator and the discrete series. Ann. Math.96 (1972) 1-30. Zbl0249.22003MR318398
  12. [12] R. Parthasarathy: A generalization of the Enright-Varadarajan modules. Comp. Math.36 (1978) 53-73. Zbl0384.17005MR515037
  13. [13] R. Parthasarathy: Criteria for the unitarizability of some highest weight modules. Proc. Indian Acad. Sci.89 (1980) 1-24.. Zbl0434.22011MR573381
  14. [14] K.R. Parthasarathy, R. Ranga Rao and V.S. Varadarajan: Representations of complex semi-simple Lie groups and Lie algebras. Ann. Math.85 (1967) 383-429. Zbl0177.18004MR225936
  15. [15] B. Speh: Unitary representations of GL(n, R) with non-trivial (g, K ) cohomology. Invent. Math.71 (1983) 443-465. Zbl0505.22015MR695900
  16. [16] B. Speh: Unitary representations of SL(n, R) and the cohomology of congruence subgroups, In. Noncommutative Harmonic Analysis and Lie Groups, Lecture Notes in Mathematics880, Springer-Verlag, BerlinHeidelbergNew York (1981). Zbl0516.22008MR644844
  17. [17] B. Speh and D. Vogan: Reducibility of generalized principal series representations. Acta Math.145 (1980) 227-299. Zbl0457.22011MR590291
  18. [18] D. Vogan: The algebraic structure of the representations of semi-simple Lie groups I. Ann. Math.109 (!979) 1-60. Zbl0424.22010MR519352
  19. [19] D. Vogan: Representations of real reductive Lie groups, Birkhäuser, Boston-Vasel-Stuttgart (1981). Zbl0469.22012MR632407
  20. [20] G. Warner: Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, BerlinHeidelbergNew York (1972). Zbl0265.22020MR498999
  21. [21] J. Rawnsley, W. Schmid and J. Wolf: Singular unitary representations and indefinite harmonic theory, to appear in J. Func. Anal., 1983. Zbl0511.22005MR699229

Citations in EuDML Documents

  1. Siddhartha Sahi, The Capelli identity and unitary representations
  2. Shingo Murakami, Vanishing theorems on cohomology associated to hermitian symmetric spaces
  3. Susana Salamanca Riba, On the unitary dual of some classical Lie groups
  4. Jens Franke, Harmonic analysis in weighted L 2 -spaces
  5. Steven Zucker, L 2 -cohomology and intersection homology of locally symmetric varieties, II
  6. Joachim Schwermer, On arithmetic quotients of the Siegel upper half space of degree two
  7. Laurent Clozel, Progrès récents vers la classification du dual unitaire des groupes réductifs réels
  8. Laurent Clozel, Patrick Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II
  9. Jürgen Rohlfs, Birgit Speh, Representations with cohomology in the discrete spectrum of subgroups of SO ( n , 1 ) ( Z ) and Lefschetz numbers
  10. Hisayosi Matumoto, On the representations of U ( m , n ) unitarily induced from derived functor modules

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