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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)
     

An external approach to unitary representations

Author(s): Marko Tadi\'c
Journal: Bull. Amer. Math. Soc. 28 (1993), 215-252.
MSC (1991): Primary
MathSciNet review: 1181278
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References:

[A$^^{\text {\bf sur}}$]
J. Arthur, \emph{Automorphic representations and number theory}, Amer. Math. Soc., Providence, RI, 1981 pp.~3--51. MR 670091
[Bb$^^{\text {\bf oth}}$]
D. Barbasch, The unitary dual for complex classical groups, Invent. Math. \textbf{96} (1989), 103--176. MR 981739
[Bg$^^{\text {\bf his}}$]
V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. \textbf{48} (1947), 568--640. MR 21942
[Bn1$^^{\text {\bf oth}}$]
J. Bernstein, All reductive $p$-adic groups are tame, Funct. Anal. Appl. \textbf{8} (1974), 91--93. MR 348045
[Bn2$^^{\text {\bf GL (n)}}$]
J. Bernstein, \emph{P-invariant distributions on GL\<$(N)$ and the classification of unitary representations of GL\<$(N)$ (nonarchimedean case)}, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, 1984 pp.~50--102. MR 748505
[BnZe1$^^{\text {\bf GL (n)}}$]
J. Bernstein and A. V. Zelevinsky, Representations of the group $GL (n,F)$, where $F$ is a local nonarchimedean field, Uspekhi Mat. Nauk. \textbf{31} (1976), 5--70. MR 425030
[BnZe2$^^{\text {\bf GL (n)}}$]
J. Bernstein and A. V. Zelevinsky, Induced representations of reductive $p$-adic groups. \RM I, Ann. Sci. \'{E}cole Norm. Sup. (4) \textbf{10} (1977), 441--472. MR 579172
[Bl$^^{\text {\bf gen}}$]
A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 251042
[BlWh$^^{\text {\bf gen}}$]
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton Univ. Press, Princeton, NJ, 1980. MR 554917
[Bu1$^^{\text {\bf gen}}$]
N. Bourbaki, \emph{Groupes de Lie r\'eels compacts}, Masson, Paris, 1982. MR 682756
[Bu2$^^{\text {\bf gen}}$]
N. Bourbaki, \emph{Mesure de Haar}, chapter 7, Hermann, Paris, 1963. MR 179291
[Cy$^^{\text {\bf oth}}$]
H. Carayol, Repr\'esentations cuspidales du groupe lin\'eaire, Ann. Sci. \'{E}cole Norm. Sup (4) \textbf{17} (1984), 191--226. MR 760676
[Ct$^^{\text {\bf sur}}$]
P. Cartier, \emph{Representations of $p$-adic groups; a survey}, Amer. Math. Soc., Providence, RI, 1979 pp.~111--155. MR 546593
[Cs$^^{\text {\bf gen}}$]
W. Casselman, Introduction to the theory of admissible representations of $p$-adic reductive groups, preprint. MR
[CsMi$^^{\text {\bf oth}}$]
W. Casselman and D. Mili\v {c}i\'{c}, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. \textbf{49} (1982), 869--930. MR 683007
[$\CL ^^{\text {\bf sur}}$]
L. Clozel, Progr\`{e}s r\'ecents vers la classification du dual unitaire des groupes r\'eductifs r\'eels, S\'eminaire Bourbaki, no. 681 (1987); Ast\'erisque {\bf 152--153} (1987), 229--252. MR 936857
[DeKaVi$^^{\text {\bf Lan}}$]
P. Deligne, D. Kazhdan, and M.-F. Vign\'eras, \emph{Repr\'esentations des alg\`ebres centrales simples $p$-adiques}, Hermann, Paris, 1984. MR 771672
[Di$^^{\text {\bf gen}}$]
J. Dixmier, Les $C^\ast $-algebras et leurs repr\'esentations, Gauthiers-Villars, Paris, 1969. MR 246136
[Du1$^^{\text {\bf sur}}$]
M. Duflo, Repr\'esentations de carr\'e int\'egrable des groupes semi-simples r\'eels, S\'eminaire Bourbaki, no. 508 (1977-1978), Lecture Notes in Math., vol. 710, Springer-Verlag, Berlin, 1979. MR 554213
[Du2$^^{\text {\bf oth}}$]
M. Duflo, Th\'eorie de Mackey pour les groupes alg\`ebriques, Acta Math. \textbf{149} (1982), 153--213. MR 688348
[Fe$^^{\text {\bf oth}}$]
J. M. G. Fell, Nonunitary dual space of groups, Acta Math. \textbf{114} (1965), 267--310. MR 186754
[Gb1$^^{\text {\bf sur}}$]
S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., vol. 83, Princeton Univ. Press, Princeton, NJ, 1975. MR 379375
[Gb2$^^{\text {\bf sur}}$]
S. Gelbart, Elliptic curves and automorphic representations, Adv. in Math. \textbf{21} (1976), 235--292. MR 439754
[Gb3$^^{\text {\bf sur}}$]
S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. \textbf{10} (1984), 177--219. MR 733692
[GfGr$^^{\text {\bf his}}$]
I. M. Gelfand and M. I. Graev, Unitary representations of the real unimodular group, Izv. Akad. Nauk SSSR Ser. Mat. \textbf{17} (1953), 189--248. MR 57261
[GfGrPi$^^{\text {\bf gen}}$]
I. M. Gelfand, M. Graev, and Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, PA, 1969. MR 233772
[GfKa$^^{\text {\bf GL (n)}}$]
I. M. Gelfand and D. A. Kazhdan, \emph{Representations of $GL (n,k)$}, Halstead Press, Budapest, 1974 pp.~95--118. MR
[GfN1$^^{\text {\bf his}}$]
I. M. Gelfand and M. A. Naimark, Unitary representations of the Lorentz group, Izv. Akad. Nauk SSSR Ser. Mat. \textbf{11} (1947), 411--504 \afterall (Russian). MR 24440
[GfN2$^^{\text {\bf his}}$]
I. M. Gelfand and M. A. Naimark, Unit\"are Darstellungen der Klassischen Gruppen {\rm (German translation of Russian publication from 1950)}, Akademie Verlag, Berlin, 1957. MR
[GfRa$^^{\text {\bf his}}$]
I. M. Gelfand and D. A. Raikov, Irreducible unitary representations of locally compact groups, Mat. Sb. \textbf{13 {\rm (55)}} (1943), 301--316. MR 11308
[Ha1$^^{\text {\bf sur}}$]
Harish-Chandra, Harmonic analysis on semisimple Lie groups, Bull. Amer. Math. Soc. \textbf{76} (1970), 529--551. MR 257282
[Ha2$^^{\text {\bf sur}}$]
Harish-Chandra, \emph{Harmonic analysis on reductive $p$-adic groups}, Amer. Math. Soc., Providence, RI, 1973 pp.~167--192. MR 340486
[Ha3$^^{\text {\bf gen}}$]
Harish-Chandra, Collected papers, Springer-Verlag, Berlin, 1983. MR
[He$^^{\text {\bf Lan}}$]
G. Henniart, On the local Langlands conjecture for $GL (n)$\RM : the cyclic case, Ann. of Math. (2) \textbf{123} (1986), 145--203. MR 825841
[Ho$^^{\text {\bf oth}}$]
R. Howe, Tamely ramified supercuspidal representations of $GL _n$, Pacific J. Math. \textbf{73} (1977), 437--460. MR 492087
[HoMr$^^{\text {\bf oth}}$]
R. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. \textbf{32} (1979), 72--96. MR 533220
[Jc1$^^{\text {\bf GL (n)}}$]
H. Jacquet, \emph{Generic representations}, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1977 pp.~91--101. MR 499005
[Jc2$^^{\text {\bf GL (n)}}$]
H. Jacquet, \emph{On the residual spectrum of $GL (n)$}, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, 1984 pp.~185--208. MR 748508
[JcL$^^{\text {\bf Lan}}$]
H. Jacquet and R. P. Langlands, Automorphic forms on $GL (2)$, Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin, 1970. MR 401654
[Jn$^^{\text {\bf oth}}$]
C. Jantzen, Degenerate principal series for symplectic groups, Mem. Amer. Math. Society, no. 488, Amer. Math. Soc., Providence, RI, 1993. MR 1134591
[Ka$^^{\text {\bf oth}}$]
D. A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. \textbf{1} (1967), 63--65. MR 209390
[Ki1$^^{\text {\bf GL (n)}}$]
A. A. Kirillov, Infinite dimensional representations of the general linear group, Dokl. Akad. Nauk SSSR \textbf{114} (1962), 37--39. MR 139691
[Ki2$^^{\text {\bf gen}}$]
A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, New York, 1976. MR 412321
[Kn$^^{\text {\bf gen}}$]
A. W. Knapp, Representation theory of semisimple groups, Princeton Univ. Press, Princeton, NJ, 1986. MR 855239
[KnZu$^^{\text {\bf sur}}$]
A. W. Knapp and G. J. Zuckerman, \emph{Classification theorems for representations of semisimple Lie groups}, Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1977 pp.~138--159. MR 476923
[KuMy$^^{\text {\bf Lan}}$]
P. Kutzko and A. Moy, On the local Langlands conjecture in prime dimension, Ann. of Math. (2) \textbf{121} (1985), 495--517. MR 794371
[L1$^^{\text {\bf sur}}$]
R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, 1970, pp.~18--86. MR 302614
[L2$^^{\text {\bf gen}}$]
R. P. Langlands, \emph{On the classification of irreducible representations of real algebraic groups}, Amer. Math. Soc., Providence, RI, 1989. MR 1011897
[Ma$^^{\text {\bf his}}$]
F. Mautner, Spherical functions over $p$-adic fields. \RM I, Amer. J. Math. \textbf{80} (1958), 441--457. MR 93558
[Mi$^^{\text {\bf oth}}$]
D. Mili\v {c}i\'{c}, On $C^\ast $-algebras with bounded trace, Glasnik Mat. \textbf{8} (1973), 7--21. MR 324429
[M }Wd$^{^{\text {\bf oth}}$]
C. M{\oe }glin and J.-L. Waldspurger, Le spectre residuel de $GL (n)$, Ann. Sci. \'{E}cole Norm. Sup. (4) \textbf{22} (1989), 605--674. MR 1026752
[My$^^{\text {\bf Lan}}$]
A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. \textbf{108} (1986), 863--930. MR 853218
[Ro$^^{\text {\bf sur}}$]
F. Rodier, Repr\'esentations de $GL (n,k)$ o\`u $k$ est un corps $p$-adique, S\'eminaire Bourbaki, no. 587 (1982), Ast\'erisque \textbf{92--93} (1982), 201--218. MR 689531
[Sh$^^{\text {\bf GL (n)}}$]
S. Sahi, On Kirillov's conjecture for archimedean fields, Compositio Math. \textbf{72} (1989), 67--86. MR 1026329
[SlTd$^^{\text {\bf oth}}$]
P. J. Sally and M. Tadi\'{c}, Induced representations and classifications for $GSp(2, F)$ and $Sp(2,F)$, M\'em. Soc. Math. France \textbf{52} (1993). MR
[Sd1$^^{\text {\bf oth}}$]
F. Shahidi, Fourier transforms of intertwining operators and Plancherel measure for $GL (n)$, Amer. J. Math. \textbf{106} (1984), 67--111. MR 729755
[Sd2$^^{\text {\bf oth}}$]
F. Shahidi, A proof of Langlands conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) \textbf{132} (1990), 273--330. MR 1070599
[Si1$^^{\text {\bf oth}}$]
A. Silberger, The Langlands quotient theorem for $p$-adic groups, Math. Ann. \textbf{236} (1978), 95--104. MR 507262
[Si2$^^{\text {\bf gen}}$]
A. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Princeton Univ. Press, Princeton, NJ, 1979. MR 544991
[Sp1$^^{\text {\bf oth}}$]
B. Speh, The unitary dual of $GL (3,\Bbb R)$ and $GL (4,\Bbb R)$, Math. Ann. \textbf{258} (1981), 113--133. MR 641819
[Sp2$^^{\text {\bf GL (n)}}$]
B. Speh, Unitary representations of $GL (n,\Bbb R)$ with nontrivial $(\germ g ,K)$-cohomology, Invent. Math. \textbf{71} (1983), 443--465. MR 695900
[St$^^{\text {\bf GL (n)}}$]
E. M. Stein, Analysis in matrix spaces and some new representations of $\SL (N,\Bbb C)$, Ann. of Math. \textbf{86} (1967), 461--490. MR 219670
[Td1$^^{\text {\bf sur}}$]
M. Tadi\'{c}, Unitary dual of $p$-adic $GL (n)$, proof of Bernstein conjectures, Bull. Amer. Math. Soc. \textbf{13} (1985), 39--42. MR 788387
[Td2$^^{\text {\bf GL (n)}}$]
M. Tadi\'{c}, Unitary representations of general linear group over real and complex field, preprint MPI/SFB 85-22 Bonn, 1985. MR
[Td3$^^{\text {\bf GL(n)}}$]
M. Tadi\'{c}, Classification of unitary representations in irreducible representations of general linear group (nonarchimedean case), Ann. Sci. \'{E}cole Norm. Sup. (4) \textbf{19} (1986), 335--382. MR 870688
[Td4$^^{\text {\bf oth}}$]
M. Tadi\'{c}, Topology of unitary dual of nonarchimedean $GL (n)$, Duke Math. J. \textbf{55} (1987), 385--422. MR 894588
[Td5$^^{\text {\bf oth}}$]
M. Tadi\'{c}, On limits of characters of irreducible unitary representations, Glasnik Mat. \textbf{23} (1988), 15--25. MR 976070
[Td6$^^{\text {\bf oth}}$]
M. Tadi\'{c}, Geometry of dual spaces of reductive groups \RM (nonarchimedean case\/\RM ), J. Analyse Math. \textbf{51} (1988), 139--181. MR 963153
[Td7$^^{\text {\bf oth}}$]
M. Tadi\'{c}, Induced representations of $GL (n,A)$ for $p$-adic division algebras $A$, J. Reine Angew. Math. \textbf{405} (1990), 48--77. MR 1040995
[Td8$^^{\text {\bf oth}}$]
M. Tadi\'{c}, \emph{On Jacquet modules of induced representations of $p$-adic symplectic groups}, Proceedings, Bowdoin College 1989, Progr. Math., vol. 101, Birkh\"auser, Boston, MA, 1991 pp.~305--314. MR 1168490
[Vo1$^^{\text {\bf gen}}$]
D. A. Vogan, Representations of real reductive groups, Birkh\"{a}user, Boston, MA, 1981. MR
[Vo2$^^{\text {\bf oth}}$]
D. A. Vogan, Unitarizability of certain series of representations, Ann. of Math. (2) \textbf{120} (1984), 141--187. MR 750719
[Vo3$^^{\text {\bf GL(n)}}$]
D. A. Vogan, The unitary dual of $GL (n)$ over an archimedean field, Invent. Math. \textbf{82} (1986), 449--505. MR 827363
[Vo4$^^{\text {\bf oth}}$]
D. A. Vogan, Unitary representations of reductive Lie groups, Princeton Univ. Press, Princeton, NJ, 1987. MR 908078
[Wh$^^{\text {\bf gen}}$]
N. Wallach, Real reductive groups, \RM {vol. 1}, Academic Press, New York, 1988. MR 929683
[Wr$^^{\text {\bf gen}}$]
G. Warner, Harmonic analysis on semi-simple Lie groups. {\rm I, II}, Springer-Verlag, Berlin, 1972. MR 498999
[We$^^{\text {\bf gen}}$]
A. Weil, Basic number theory, Springer-Verlag, New York, 1974. MR 427267
[Ze$^^{\text {\bf GL(n)}}$]
A. V. Zelevinsky, Induced representations of reductive $p$-adic groups {\rm II}, On irreducible representations of $GL (n)$, Ann. Sci. \'{E}cole Norm. Sup. (4) \textbf{13} (1980), 165--210. MR 584084

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DOI: 10.1090/S0273-0979-1993-00372-0
PII: S 0273-0979(1993)00372-0