Book Review
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Book Information:
Author:
Jr. A. W. Knapp and D. A. Vogan
Title:
Cohomological induction and unitary representations
Additional book information:
Princeton Univ. Press,
Princeton, NJ,
1995,
xvii +948 pp.,
ISBN 0-691-03756-6
Dan Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. MR 981739, DOI 10.1007/BF01393972
Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
[BZ] L. Barchini and R. Zierau, Square integrable harmonic forms and representation theory, Duke Math. J. 92, no. 3 (1998), 645-664. CMP 98:12
T. J. Enright and N. R. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), no. 1, 1–15. MR 563362
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Generalized Functions, vol. 6, Academic Press, Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition. MR 1071179
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
S. Kumaresan, On the canonical $k$-types in the irreducible unitary $g$-modules with nonzero relative cohomology, Invent. Math. 59 (1980), no. 1, 1–11. MR 575078, DOI 10.1007/BF01390311
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Henryk Hecht, Dragan Miličić, Wilfried Schmid, and Joseph A. Wolf, Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), no. 2, 297–332. MR 910203, DOI 10.1007/BF01388707
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 1, Mathematics and its Applications (Soviet Series), vol. 72, Kluwer Academic Publishers Group, Dordrecht, 1991. Simplest Lie groups, special functions and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1143783, DOI 10.1007/978-94-011-3538-2
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
[MP] D. Milicic and P. Pandzic, Equivariant derived categories, Zuckerman functors and localization, Geometry and representation theory of real and -adic groups, Progress in Mathematics, vol. 158, Birkhäuser-Boston, 1997, pp. 209-242. CMP 98:05
Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 286942, DOI 10.2307/1970751
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. MR 750719, DOI 10.2307/2007074
David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
Hon-Wai Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal. 129 (1995), no. 2, 428–454. MR 1327186, DOI 10.1006/jfan.1995.1058
- [Ba]
- D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), 103-176.MR 90c:22044
- [BL]
- J. Bernstein and V. Lunts, Equivariant sheaves and functors, LNM, vol. 1578, Springer-Verlag, Berlin-Heidelberg-Tokyo, 1994. MR 95k:55012
- [BZ]
- L. Barchini and R. Zierau, Square integrable harmonic forms and representation theory, Duke Math. J. 92, no. 3 (1998), 645-664. CMP 98:12
- [EW]
- T. Enright and N. Wallach, Notes on cohomological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), 1-15. MR 81c:17013
- [BW]
- A. Borel and N. Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematics Studies, no. 94, 1980, Princeton University Press and University of Tokyo Press. MR 83c:22018
- [GGPS]
- I. M. Gelfand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, 1990, Academic Press. MR 91g:11052
- [K]
- A. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, 1986, Princeton University Press. MR 87j:22022
- [Ku]
- S. Kumaresan, On the canonical -types in the irreducible unitary -modules with non-zero relative cohomology, Invent. Math. 59 (1980), 1-11. MR 83c:17011
- [H]
- S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, 1978. MR 80k:53081
- [HMSW]
- H. Hecht, D. Milicic, W. Schmid, and J. Wolf, Localization and standard modules for real semisimple groups. I, Invent. Math. 90 (1987), 297-332. MR 89e:22025
- [KV]
- A. V. Klimyk and N. A. Vilenkin, Representations of Lie groups and special functions, Kluwer Academic Publishers, Dordrecht, Boston, 1991. MR 93h:33009
- [M]
- S. MacLane, Homology, Academic Press-Springer Verlag, New York, Berlin, Göttingen, 1963. MR 28:122
- [MP]
- D. Milicic and P. Pandzic, Equivariant derived categories, Zuckerman functors and localization, Geometry and representation theory of real and -adic groups, Progress in Mathematics, vol. 158, Birkhäuser-Boston, 1997, pp. 209-242. CMP 98:05
- [S]
- W. Schmid, On a conjecture of Langlands, Ann. of Math. 93 (1971), 1-43. MR 44:4149
- [V1]
- D. Vogan, Representations of real reductive groups, Progress in Mathematics, 1981, Birkhäuser, Boston-Basel-Stuttgart. MR 83c:22022
- [V2]
- D. Vogan, Unitarizability of certain series of representations, Ann. Math. 120 (1984), 141-187. MR 86h:22028
- [VZ]
- D. Vogan and G. Zuckerman, Unitary representations with nonzero cohomology, Comp. Math. 53 (1984), 51-90. MR 86k:22040
- [W]
- N. Wallach, Real reductive groups, I, II, Academic Press, Boston, 1988, 1992. MR 89i:22029; MR 93m:22018
- [Wo]
- H. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal. 129, no. 2 (1995), 428-454. MR 96c:22024
Review Information:
Reviewer:
Dan Barbasch
Affiliation:
Cornell University
Email:
barbasch@math.cornell.edu
Journal:
Bull. Amer. Math. Soc.
36 (1999), 391-397
DOI:
https://doi.org/10.1090/S0273-0979-99-00782-X
Published electronically:
April 21, 1999
Review copyright:
© Copyright 1999
American Mathematical Society