On the representations of U ( m , n ) unitarily induced from derived functor modules

Hisayosi Matumoto

Compositio Mathematica (1996)

  • Volume: 100, Issue: 1, page 1-39
  • ISSN: 0010-437X

How to cite

top

Matumoto, Hisayosi. "On the representations of $U(m, n)$ unitarily induced from derived functor modules." Compositio Mathematica 100.1 (1996): 1-39. <http://eudml.org/doc/90420>.

@article{Matumoto1996,
author = {Matumoto, Hisayosi},
journal = {Compositio Mathematica},
keywords = {cohomological induction; unitary representations; representations; indefinite unitary groups; derived functor modules; unitary degenerate series},
language = {eng},
number = {1},
pages = {1-39},
publisher = {Kluwer Academic Publishers},
title = {On the representations of $U(m, n)$ unitarily induced from derived functor modules},
url = {http://eudml.org/doc/90420},
volume = {100},
year = {1996},
}

TY - JOUR
AU - Matumoto, Hisayosi
TI - On the representations of $U(m, n)$ unitarily induced from derived functor modules
JO - Compositio Mathematica
PY - 1996
PB - Kluwer Academic Publishers
VL - 100
IS - 1
SP - 1
EP - 39
LA - eng
KW - cohomological induction; unitary representations; representations; indefinite unitary groups; derived functor modules; unitary degenerate series
UR - http://eudml.org/doc/90420
ER -

References

top
  1. 1 Barbasch, D.: Unipotent representations and unitarity, in: Non-Commutative Harmonic Analysis and Lie Groups, Springer Lecture Notes in Mathematics vol. 1243, Springer-Verlag, Berlin, Heidelberg, New York, 1987, 73-85. Zbl0629.22007MR897538
  2. 2 Barbasch, D. and Vogan Jr., D.A.: Weyl group representations and nilpotent orbits, in: P. C. Trombi, editor, 'Representation Theory of Reductive Groups', Progress in Mathematics Vol. 40, 21-33, Birkhäuser, Boston-Basel- Stuttgart, 1983. Zbl0537.22013MR733804
  3. 3 Barbasch, D. and Vogan Jr., D.A.: Unipotent representations of complex semisimple Lie groups, Ann. of Math.121 (1985), 41-110. Zbl0582.22007MR782556
  4. 4 Bien, F.: 'D-modules and Spherical Representations', Math. Notes vol. 39, Princeton University Press, Princeton, NJ, 1990. Zbl0723.22014MR1082342
  5. 5 Borho, W. and Brylinski, J.- L.: Differential operators on homogeneous spaces III, Invent. Math.80 (1985), 1-68. Zbl0577.22014MR784528
  6. 6 Borho, W. and Jantzen, J.C.: Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra, Invent. Math.39 (1977), 1-53. Zbl0327.17002MR453826
  7. 7 Chang, J.T.: Special K-types, tempered characters and the Beilinson-Bemstein realization, Duke Math. J.56 (1988), 345-383. Zbl0655.22010MR932850
  8. 8 Dixmier, J.: 'Enveloping Algebras'North-Holland, Amsterdam, New York, Oxford, 1977. Zbl0339.17007MR498740
  9. 9 Enright, T.J.: Relative Lie algebra cohomology and unitary representations of complex Lie groups, Duke Math. J.46 (1979), 513-525. Zbl0427.22010MR544243
  10. 10 Gross, K.I.: The dual of parabolic subgroup and a degenerate principal series of Sp(n, C), Amer. J. Math.93 (1971) 398-428. Zbl0229.22025MR304558
  11. 11 Hecht, H., Miličič, D., Schmid, W., and Wolf, J.: Localization and standard modules for real semisimple Lie groups I: The duality theorem, Invent. Math.90 (1987), 297-332. Zbl0699.22022MR910203
  12. 12 Howe, R. and Tan, E.: Homogeneous functions on light cones: The infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc.28 (1993), 1-74. Zbl0794.22012MR1172839
  13. 13 Jantzen, J.C.: 'Einhüllende Algebren halbeinfacher Lie-Algebren' Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Zbl0541.17001MR721170
  14. 14 Johnson, K.D.: Degenerate principal series and compact groups, Maths. Ann.287 (1990), 703-718. Zbl0687.22003MR1066825
  15. 15 Kashiwara, M. and Vergne, M.: Functions on the Shilov boundary of the generalized half-plane, in: Non-Commutative Harmonic Analysis, Lecture Notes in Mathematics No. 728, Springer-Verlag, Berlin-Heidelberg -New York, 1979. Zbl0416.22006MR548329
  16. 16 Knapp, A.W.: 'Representation Theory of Semisimple Groups, An Overview Based on Examples'Princeton Mathematical series, 36, Princeton University Press, Lawrenceville, New Jersey, 1986. Zbl0604.22001MR855239
  17. 17 Knapp, A.W.: 'Lie Groups, Lie Algebras, and Cohomology' Mathematical Notes34, Princeton University Press, Princeton, New Jersey, 1988. Zbl0648.22010MR938524
  18. 18 Knapp, A.W. and Zuckerman, G.J.: Classification of irreducible tempered representations of semisimple Lie groups, Ann. of Math.116 (1982), 389-455. Zbl0516.22011MR672840
  19. 19 Kobayashi, T.: Singular unitary representations and discrete series for indefinite Stiefel manifolds U(p, q; F)/U(p - m, q; F), Memoirs of AMS vol. 462, 1992. Zbl0752.22007
  20. 20 Matsuki, T.: A description of discrete series for semisimple symmetric spaces II, Adv. Stud. in Pure Math. vol. 14, Kinokuniya Book Store and North-Holland, 1988, 531-540. Zbl0719.22003MR1039851
  21. 21 Matumoto, H.: Cohomological Hardy space for SU(2, 2), Adv. Stud. in Pure Math. vol. 14, Kinokuniya Book Store and North-Holland, 1988, 469-497. Zbl0725.22006MR1039848
  22. 22 Miličič, D.: Intertwining functors and irreducibility of standard Harish-Chandra sheaves, in: Harmonic Analysis on Reductive Groups, Progress in Mathematics vol. 101, Birkhäuser, Boston, Basel, Berlin, 1991, 209-222. Zbl0760.22019MR1168485
  23. 23 Mirkovic, I.: Classification of irreducible tempered representations of semisimple Lie groups, Ph.D. Thesis, Univ. of Utah, Salt Lake City, 1986. 
  24. 24 Salamanca Riba, S.: On the unitary dual of some classical Lie groups, Composito Math.68 (1988), 251-303. Zbl0692.22007MR971329
  25. 25 Sahi, S.: The Capelli identity and unitary representations, Compositio Math.81 (1992), 247-260. Zbl0758.22008MR1149169
  26. 26 Sahi, S.: Unitary representations on the Shilov boundary of a symetric tube domain, in: Representation Theory of Groups and Algebras, Contemporary Mathematics, vol. 145, American Mathematical Society, 1993, 275-286. Zbl0790.22010MR1216195
  27. 27 Schmid, W.: On the characters of the discrete series (the Hermitian symmetric case), Invent. Math.30 (1975), 47-144. Zbl0324.22007MR396854
  28. 28 Schmid, W.: Two character identities for semisimple Lie groups, in: Non-Commutative Harmonic Analysis, Springer Lecture Notes in Mathematics vol. 587, 1977, 196-225. Zbl0362.22015MR507247
  29. 29 Schmid, W.: Geometric constructions of representations, Adv. Stud. in Pure Math. vol. 14, Kinokuniya Book Store, 349-368,1988. Zbl0706.22012MR1039843
  30. 30 Schmid, W.: Construction and classification of irreducible Harish-Chandra modules, in: Harmonic Analysis on Reductive Groups, Progress in Mathematics vol. 101, Birkhäuser, Boston, Basel, Berlin, 1991, 235-275. Zbl0751.22003MR1168487
  31. 31 Schmid, W. and Wolf, J.A.: Geometric quantization and derived functor modules for semisimple Lie group, J. of Funct. Anal.90 (1990), 48-112. Zbl0781.22009MR1047577
  32. 32 Soergel, W.: Universelle versus relative Einhüllende: Eine geometrische Untersuchung von Quotientienten von universellen Einhüllenden halbeinfacher Lie-Algebren, Dissertation, Universität Hamburg, Hamburg, 1987. Zbl0649.17012
  33. 33 Speh, B.: Degenerate series representations of the universal covering group of SU(2, 2), J. Funct. Anal.33 (1979), 95-118. Zbl0415.22012MR545386
  34. 34 Speh, B. and Vogan Jr., D.A.: Reductibility of generalized series representations, Acta Math. 145 (1980), 227-299. Zbl0457.22011MR590291
  35. 35 VoganJr., D.A.: Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math.48 (1978), 75-98. Zbl0389.17002MR506503
  36. 36 Vogan Jr., D.A.: 'Representations of Real reductive Lie Groups', Progress in Mathematics, Birkhäuser, 1982. Zbl0469.22012
  37. 37 Vogan Jr., D.A.: Irreducible characters of semisimple Lie groups III, Invent. Math.71 (1983), 381-417. Zbl0505.22016MR689650
  38. 38 Vogan Jr., D.A.: Unitarizability of certain series of representations, Ann. of Math.120 (1984), 141-187. Zbl0561.22010MR750719
  39. 39 Vogan Jr., D.A.: 'Unitary Representations of reductive Lie Groups', Annuals of Mathematics Studies, Princeton University Press, 1987. Zbl0626.22011MR908078
  40. 40 Vogan Jr., D.A.: The unitary dual of GL(n) over an archimedean field, Invent. Math.83 (1986), 449-505. Zbl0598.22008MR827363
  41. 41 Vogan Jr., D.A.: Irreducibilities of discrete series representations for semisimple symmetric spaces, Adv. Stud. in Pure Math. vol. 14, Kinokuniya Book Store, 1988, 381-417. Zbl0733.22008
  42. 42 Vogan Jr., D.A.: Dixmier algebras, sheets, and representation theory, in: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Actes du colloque en l'honneur de Jacques Dixmier, Progress in Mathmatics, vol 92, Birkhäuser, Basel, Boston, 1990, 333-395. Zbl0854.17010MR1103596
  43. 43 Vogan Jr., D.A. and Zuckerman, G.: Unitary representations with non-zero cohomology, Compositio Math.53 (1984), 51-90. Zbl0692.22008MR762307
  44. 44 Wallach, N.R.: On the unitarizability of derived functor modules, Invent. Math.78 (1984), 131-141. Zbl0547.22008MR762359
  45. 45 Wallach, N.R.: 'Real reductive Groups I', Academic press, 1987. Zbl0666.22002MR929683

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.