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Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case). (English) Zbl 0614.22005
The author classifies the irreducible unitary representations of the general linear group GL(n) over a local, non-Archimedean field. He constructs a set B so that every unitary representation is either in B or is unitarily induced from a representation in B. The representations in B are either irreducible quotients of representations induced from square integrable representations or complementary series representations. The classifications of the unitary dual is given in Langlands as well as in Zelevinski parameters.
A similar classification of the unitary dual for Archimedean fields was obtained by D. Vogan.
Reviewer: B.Speh

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
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References:
[1] J. N. BERNSTEIN , All Reductive p-adic Groups Are Tame (Funct. Anal. Appl., Vol. 8, 1974 , pp. 91-93). MR 50 #543 | Zbl 0298.43013 · Zbl 0298.43013 · doi:10.1007/BF01078592 · mi.mathnet.ru
[2] J. N. BERNSTEIN , P-Invariant Distributions on GL (N) and the Classification of Unitary Representations of GL (N) (Non-Archimedean Case), in Lie Group Representations II (Proceedings, University of Maryland, 1982 - 1983 , Lecture Notes in Math., Vol. 1041, Springer-Verlag, Berlin, 1984 , pp. 50-102). Zbl 0541.22009 · Zbl 0541.22009
[3] J. N. BERNSTEIN , P. DELIGNE , D. KAZHDAN and M. F. VIGNERAS , Représentations des groupes réductifs sur un corps local , Hermann, Paris, 1984 . Zbl 0544.00007 · Zbl 0544.00007
[4] J. N. BERNSTEIN and A. V. ZELEVINSKY , Representations of the Group GL (n, F), where F Is a Local Non-Archimedean Field [Uspekhi Mat. Nauk, Vol. 31, No. 3, 1976 , pp. 5-70 (= Russian Math. Surveys, Vol. 31, No. 3, 1976 , pp. 1-68)]. Zbl 0348.43007 · Zbl 0348.43007 · doi:10.1070/RM1976v031n03ABEH001532
[5] J. N. BERNSTEIN and A. V. ZELEVINSKY , Induced Representations of Reductive p-adic Groups, I (Ann. Scient. Éc. Norm. Sup., Vol. 10, 1977 , pp. 441-472). Numdam | MR 58 #28310 | Zbl 0412.22015 · Zbl 0412.22015 · numdam:ASENS_1977_4_10_4_441_0 · eudml:82002
[6] P. CARTIER , Representations of p-adic Groups : a Survey , in Proc. Sympos. Pure Math., Vol. XXXIII, part 1, Amer. Math. Soc., Providence, R. I., 1979 , pp. 111-155. MR 81e:22029 | Zbl 0421.22010 · Zbl 0421.22010
[7] W. CASSELMAN , Introduction to the Theory of Admissible Representations of p-adic Reductive Groups , preprint.
[8] G. VAN DIJK , Computation of Certain Induced Characters of p-adic Groups (Math. Ann., Vol. 199, 1972 , pp. 229-240). MR 49 #3043 | Zbl 0231.22018 · Zbl 0231.22018 · doi:10.1007/BF01429876 · eudml:162326
[9] D. FLATH , Decomposition of Representations into Tensor Products , in Proc. Sympos. Pure Math., Vol. XXXIII, part I, Amer. Math. Soc., Providence, R. I., 1979 , pp. 179-183. MR 81f:22028 | Zbl 0414.22019 · Zbl 0414.22019
[10] I. M. GELFAND and M. I. GRAEV , Representations of a Group of the Second Order with Elements from a Locally Compact Field (Russian Math. Surveys, Vol. 18, 1963 , pp. 29-100). MR 27 #5864 | Zbl 0166.40201 · Zbl 0166.40201
[11] I. M. GELFAND , M. GRAEV and I. I. PIATETSKI-SHAPIRO , Automorphic Functions and Representation Theory , W. B. Sannders Co., Philadelphia, 1969 . Zbl 0177.18003 · Zbl 0177.18003
[12] I. M. GELFAND and D. A. KAZHDAN , Representations of GL (n, K) , in Lie Groups and Their Representations, Akademiai Kiado, Budapest, 1974 , pp. 95-118. Zbl 0348.22011 · Zbl 0348.22011
[13] I. M. GELFAND and M. A. NEUMARK , Unitare Darstellungen der Klassichen Gruppen , German translation, Akademie Verlag, Berlin, 1957 . Zbl 0077.03405 · Zbl 0077.03405
[14] H. JACQUET , Generic Representations , in Non-Commutative Harmonic Analysis (Lecture Notes in Math., Vol. 587, Springer-Verlag, Berlin, 1977 , pp. 91-101). MR 58 #16985 | Zbl 0357.22010 · Zbl 0357.22010
[15] H. JACQUET , Principal L-functions of the linear group , in Proc. Sympos. Pure Math., Vol. XXXIII, part 2, Amer. Math. Soc., Providence, R. I., 1979 , pp. 63-86. MR 81f:22029 | Zbl 0413.12007 · Zbl 0413.12007
[16] H. JACQUET , On the Residual Spectrum of GL(n) , in Lie Group Representations II (Proceedings, University of Maryland, 1982 - 1983 , Lecture Notes in Math., Vol. 1041, Springer-Verlag, Berlin, 1984 , pp. 185-208). MR 85k:22045 | Zbl 0539.22016 · Zbl 0539.22016
[17] A. KNAPP and B. SPEH , Status of Classification of Irreducible Unitary Representations , in Harmonic Analysis (Proceedings, Minneapolis, 1981 , Lecture Notes in Math., Vol. 908, Springer-Verlag, Berlin, 1982 ). Zbl 0496.22018 · Zbl 0496.22018
[18] D. MILIČIĆ , On C*-Algebras with Bounded Trace (Glasnik Mat., Vol. 8, (28), 1973 , pp. 7-22). MR 48 #2781 | Zbl 0265.46072 · Zbl 0265.46072
[19] C. MOEGLIN and J.-L. WALDSPURGER , Sur l’involution de Zelevinsky , preprint, Paris, 1985 . · Zbl 0594.22008
[20] G. I. OLSHANSKY , Intertwining Operators and Complementary Series in the Class of Representations of the General Group of Matrices Over a Locally Compact Division Algebra, Induced from Parabolic Subgroups (Mat. Sb., Vol. 93, No. 2, 1974 , pp. 218-253). · Zbl 0298.22016
[21] F. RODIER , Représentations de GL (n, k) où k est un corps p-adique [Séminaire Bourbaki, n^\circ 587, 1982 , (Astérisque, 92-93, 1982 , pp. 201-218)]. Numdam | MR 84h:22040 | Zbl 0506.22019 · Zbl 0506.22019 · numdam:SB_1981-1982__24__201_0 · eudml:109988
[22] J. ROGAWSKI , Representations of GL (n) and Division Algebras Over a p-adic Field (Duke Math. J., Vol. 50, 1983 , pp. 161-196). Article | MR 84j:12018 | Zbl 0523.22015 · Zbl 0523.22015 · doi:10.1215/S0012-7094-83-05006-8 · minidml.mathdoc.fr
[23] B. SPEH , The Unitary Dual of GL (3, \Bbb R) and GL (4, \Bbb R) (Math. Ann., Vol. 258, 1981 , pp. 113-133). MR 83i:22025 | Zbl 0483.22005 · Zbl 0483.22005 · doi:10.1007/BF01450529 · eudml:163582
[24] B. SPEH , Unitary Representations of GL (n, \Bbb R) with Non-Trivial (g, K) Cohomology (Invent. Math., Vol. 71, 1983 , pp. 443-465). MR 84k:22024 | Zbl 0505.22015 · Zbl 0505.22015 · doi:10.1007/BF02095987 · eudml:142998
[25] E. M. STEIN , Analysis in Matrix Spaces and Some New Representations of SL (N, \Bbb C) (Ann. of Math., Vol. 86, 1967 , pp. 461-490). MR 36 #2749 | Zbl 0188.45303 · Zbl 0188.45303 · doi:10.2307/1970611
[26] M. TADIĆ , The C*-algebra of SL (2, k) (Glasnik Mat., Vol. 17, (37), 1982 , pp. 249-263). MR 84g:22038 | Zbl 0504.22014 · Zbl 0504.22014
[27] M. TADIĆ , The Topology of the Dual Space of a Reductive Group Over a Local Field (Glasnik Mat., 18, (38), 1983 , pp. 259-279). MR 85k:22043 | Zbl 0536.22026 · Zbl 0536.22026
[28] M. TADIĆ , Proof of a Conjecture of Bernstein , (Math. Ann., Vol. 272, 1985 , pp. 11-16). MR 87i:22049 | Zbl 0547.22010 · Zbl 0547.22010 · doi:10.1007/BF01455923 · eudml:163992
[29] M. TADIĆ , Unitary Dual of p-Adic GL (n). Proof of Bernstein Conjectures , (Bull. Amer. Math. Soc., Vol. 13, No. 1, 1985 , pp. 39-42). Article | MR 86j:22029 | Zbl 0583.22008 · Zbl 0583.22008 · doi:10.1090/S0273-0979-1985-15355-8 · minidml.mathdoc.fr
[30] M. TADIĆ , Unitary Representations of General Linear Group Over Real and Complex Field , preprint, Bonn, 1985 .
[31] D. A. VOGAN , Jr., Understanding the Unitary Dual , in Lie Group Representations I (Proceedings, University of Maryland, 1982 - 1983 , Lecture Notes in Math., Vol. 1024, Springer-Verlag, Berlin 1983 , pp. 264-286). MR 86b:22027 | Zbl 0527.22017 · Zbl 0527.22017
[32] N. R. WALLACH , Representations of Reductive Lie Groups , in Proc. Sympos. Pure Math., Vol. XXXIII, part 1, Amer. Math. Soc., Providence, R. I., 1979 , pp. 71-86). MR 80m:22024 | Zbl 0421.22006 · Zbl 0421.22006
[33] A. V. ZELEVINSKY , Induced representations of reductive p-adic groups II (Ann. Scient. Éc. Norm. Sup., Vol. 13, 1980 , pp. 165-210). Numdam | MR 83g:22012 | Zbl 0441.22014 · Zbl 0441.22014 · numdam:ASENS_1980_4_13_2_165_0 · eudml:82048
[34] A. V. ZELEVINSKY , p-adic analogue of the Kazhdan-Lusztig conjecture (Funct. Anal. Appl., Vol. 15, 1981 , pp. 83-92). MR 84g:22039 | Zbl 0476.22014 · Zbl 0476.22014 · doi:10.1007/BF01082279
[35] J.-L. WALDSPURGER , Algèbres de Hecke et induites de représentations cuspidales, pour GL (N) (Journal für die Reine und angewandte Matematik, No. 370, 1986 , pp. 127-191). Article | MR 87m:22048 | Zbl 0586.20020 · Zbl 0586.20020 · doi:10.1515/crll.1986.370.127 · crelle:GDZPPN00220388X · eudml:152864
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