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Bulletin of the American Mathematical Society

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An external approach to unitary representations
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Bull. Amer. Math. Soc. 28 (1993), 215-252 Request permission
References
  • James Arthur, Automorphic representations and number theory, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 3–51. MR 670091
  • Dan Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. MR 981739, DOI 10.1007/BF01393972
  • V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 21942, DOI 10.2307/1969129
  • I. N. Bernšteĭn, All reductive ${\mathfrak {p}}$-adic groups are of type I, Funkcional. Anal. i Priložen. 8 (1974), no. 2, 3–6 (Russian). MR 0348045
  • —, P-invariant distributions on ${GL(N)}$ and the classification of unitary representations of ${GL(N)}$ (nonarchimedean case), Lie Group Representations II, Proceedings, Univ. Maryland 1982-83, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, 1984, pp. 50-102.
  • I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
  • I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
  • Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
  • 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
  • Nicolas Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1982 (French). Chapitre 9. Groupes de Lie réels compacts. [Chapter 9. Compact real Lie groups]. MR 682756
  • —, Mesure de Haar, Intégration, chapter 7, Hermann, Paris, 1963.
  • H. Carayol, Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 2, 191–225 (French). MR 760676, DOI 10.24033/asens.1470
  • P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
  • W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
  • William Casselman and Dragan Miličić, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), no. 4, 869–930. MR 683007, DOI 10.1215/S0012-7094-82-04943-2
  • Laurent Clozel, Progrès récents vers la classification du dual unitaire des groupes réductifs réels, Astérisque 152-153 (1987), 5, 229–252 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR 936857
  • P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples $p$-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33–117 (French). MR 771672
  • J. Dixmier, Les ${C^{\ast }}$-algebras et leurs représentations, Gauthiers-Villars, Paris, 1969.
  • Michel Duflo, Représentations de carré intégrable des groupes semi-simples réels, Séminaire Bourbaki, 30e année (1977/78), Lecture Notes in Math., vol. 710, Springer, Berlin, 1979, pp. Exp. No. 508, pp. 22–40 (French). MR 554213
  • Michel Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math. 149 (1982), no. 3-4, 153–213 (French). MR 688348, DOI 10.1007/BF02392353
  • J. M. G. Fell, Non-unitary dual spaces of groups, Acta Math. 114 (1965), 267–310. MR 186754, DOI 10.1007/BF02391824
  • Stephen S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0379375, DOI 10.1515/9781400881611
  • Stephen Gelbart, Elliptic curves and automorphic representations, Advances in Math. 21 (1976), no. 3, 235–292. MR 439754, DOI 10.1016/S0001-8708(76)80001-1
  • Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219. MR 733692, DOI 10.1090/S0273-0979-1984-15237-6
  • I. M. Gel′fand and M. I. Graev, Unitary representations of the real unimodular group (principal nondegenerate series), Izvestiya Akad. Nauk SSSR. Ser. Mat. 17 (1953), 189–248 (Russian). MR 0057261
  • I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 0233772
  • I. M. Gelfand and D. A. Kazhdan, Representations of ${GL(n,k)}$, Lie Groups and Their Representations, Halstead Press, Budapest, 1974, pp. 95-118.
  • I. M. Gel′fand and M. A. Naĭmark, Unitary representations of the Lorentz group, Izvestiya Akad. Nauk SSSR. Ser. Mat. 11 (1947), 411–504 (Russian). MR 0024440
  • I. M. Gelfand and M. A. Neumark, Unitäre Darstellungen der klassischen Gruppen, Akademie-Verlag, Berlin, 1957 (German). MR 0085262
  • I. Gelfand and D. Raikov, Irreducible unitary representations of locally bicompact groups, Rec. Math. [Mat. Sbornik] N. S. 13(55) (1943), 301–316. MR 0011308
  • Harish-Chandra, Harmonic analysis on semisimple Lie groups, Bull. Amer. Math. Soc. 76 (1970), 529–551. MR 257282, DOI 10.1090/S0002-9904-1970-12442-9
  • Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 167–192. MR 0340486
  • Emmy Noether, Gesammelte Abhandlungen, Springer-Verlag, Berlin-New York, 1983 (German). Edited and with an introduction by Nathan Jacobson; With an introductory address by P. S. Alexandrov [P. S. Aleksandrov]. MR 703862
  • Guy Henniart, On the local Langlands conjecture for $\textrm {GL}(n)$: the cyclic case, Ann. of Math. (2) 123 (1986), no. 1, 145–203. MR 825841, DOI 10.2307/1971354
  • Roger E. Howe, Tamely ramified supercuspidal representations of $\textrm {Gl}_{n}$, Pacific J. Math. 73 (1977), no. 2, 437–460. MR 492087, DOI 10.2140/pjm.1977.73.437
  • Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), no. 1, 72–96. MR 533220, DOI 10.1016/0022-1236(79)90078-8
  • Hervé Jacquet, Generic representations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 91–101. MR 0499005
  • —, On the residual spectrum of ${GL(n)}$, Lie Group Representations II, Proceedings, Univ. of Maryland 1982-83, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, 1984, pp. 185-208.
  • H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
  • Chris Jantzen, Degenerate principal series for symplectic groups, Mem. Amer. Math. Soc. 102 (1993), no. 488, xiv+111. MR 1134591, DOI 10.1090/memo/0488
  • D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
  • A. A. Kirillov, Infinite-dimensional representations of the complete matrix group, Dokl. Akad. Nauk SSSR 144 (1962), 37–39 (Russian). MR 0139691
  • A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321, DOI 10.1007/978-3-642-66243-0
  • 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
  • A. W. Knapp and Gregg Zuckerman, Classification theorems for representations of semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 138–159. MR 0476923
  • Philip Kutzko and Allen Moy, On the local Langlands conjecture in prime dimension, Ann. of Math. (2) 121 (1985), no. 3, 495–517. MR 794371, DOI 10.2307/1971207
  • R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 18–61. MR 0302614
  • R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR 1011897, DOI 10.1090/surv/031/03
  • F. I. Mautner, Spherical functions over ${\mathfrak {P}}$-adic fields. I, Amer. J. Math. 80 (1958), 441–457. MR 93558, DOI 10.2307/2372794
  • Dragan Miličić, On $C^{\ast }$-algebras with bounded trace, Glasnik Mat. Ser. III 8(28) (1973), 7–22 (English, with Serbo-Croatian summary). MR 324429
  • C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
  • Allen Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), no. 4, 863–930. MR 853218, DOI 10.2307/2374518
  • François Rodier, Représentations de $\textrm {GL}(n,\,k)$ où $k$ est un corps $p$-adique, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 201–218 (French). MR 689531
  • Siddhartha Sahi, On Kirillov’s conjecture for Archimedean fields, Compositio Math. 72 (1989), no. 1, 67–86. MR 1026329
  • P. J. Sally and M. Tadić, Induced representations and classifications for ${GS_{p}(2,F)}$ and ${Sp(2,F)}$, Mém. Soc. Math. France 52 (1993).
  • Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $\textrm {GL}(n)$, Amer. J. Math. 106 (1984), no. 1, 67–111. MR 729755, DOI 10.2307/2374430
  • Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
  • Allan J. Silberger, The Langlands quotient theorem for $p$-adic groups, Math. Ann. 236 (1978), no. 2, 95–104. MR 507262, DOI 10.1007/BF01351383
  • Allan J. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991
  • Birgit Speh, The unitary dual of $\textrm {Gl}(3,\,\textbf {R})$ and $\textrm {Gl}(4,\,\textbf {R})$, Math. Ann. 258 (1981/82), no. 2, 113–133. MR 641819, DOI 10.1007/BF01450529
  • Birgit Speh, Unitary representations of $\textrm {Gl}(n,\,\textbf {R})$ with nontrivial $({\mathfrak {g}},\,K)$-cohomology, Invent. Math. 71 (1983), no. 3, 443–465. MR 695900, DOI 10.1007/BF02095987
  • E. M. Stein, Analysis in matrix spaces and some new representations of $\textrm {SL}(N,\,C)$, Ann. of Math. (2) 86 (1967), 461–490. MR 219670, DOI 10.2307/1970611
  • Marko Tadić, Unitary dual of $p$-adic $\textrm {GL}(n)$. Proof of Bernstein conjectures, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 39–42. MR 788387, DOI 10.1090/S0273-0979-1985-15355-8
  • —, Unitary representations of general linear group over real and complex field, preprint MPI/SFB 85-22 Bonn, 1985.
  • Marko Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382. MR 870688, DOI 10.24033/asens.1510
  • Marko Tadić, Topology of unitary dual of non-Archimedean $\textrm {GL}(n)$, Duke Math. J. 55 (1987), no. 2, 385–422. MR 894588, DOI 10.1215/S0012-7094-87-05522-0
  • M. Tadić, On limits of characters of irreducible unitary representations, Glas. Mat. Ser. III 23(43) (1988), no. 1, 15–25 (English, with Serbo-Croatian summary). MR 976070
  • Marko Tadić, Geometry of dual spaces of reductive groups (non-Archimedean case), J. Analyse Math. 51 (1988), 139–181. MR 963153, DOI 10.1007/BF02791122
  • Marko Tadić, Induced representations of $\textrm {GL}(n,A)$ for $p$-adic division algebras $A$, J. Reine Angew. Math. 405 (1990), 48–77. MR 1040995, DOI 10.1515/crll.1990.405.48
  • Marko Tadić, On Jacquet modules of induced representations of $p$-adic symplectic groups, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 305–314. MR 1168490
  • D. A. Vogan, Representations of real reductive groups, Birkhäuser, Boston, MA, 1981.
  • 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
  • —, The unitary dual of ${GL(n)}$ over an archimedean field, Invent. Math. 82 (1986), 449-505.
  • David A. Vogan Jr., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, vol. 118, Princeton University Press, Princeton, NJ, 1987. MR 908078
  • Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
  • Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999
  • André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 0427267, DOI 10.1007/978-3-642-61945-8
  • A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084, DOI 10.24033/asens.1379
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  • © Copyright 1993 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 28 (1993), 215-252
  • MSC: Primary 22E50
  • DOI: https://doi.org/10.1090/S0273-0979-1993-00372-0
  • MathSciNet review: 1181278