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An explicit Plancherel formula for $ {\rm U}(2,1)$


Authors: David Jabon, C. David Keys and Allen Moy
Journal: Trans. Amer. Math. Soc. 341 (1994), 157-171
MSC: Primary 22E50; Secondary 11F70
MathSciNet review: 1106191
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Abstract: The admissible duals of quasi-split unitary groups over nonarchimedean fields are determined. The set of irreducible unitarizable representations, and the Plancherel measure on the unitary dual, is given explicitly.


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  • [1] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
  • [2] Laurent Clozel, Sur une conjecture de Howe. I, Compositio Math. 56 (1985), no. 1, 87–110 (English, with French summary). MR 806844
  • [3] Laurent Clozel, Orbital integrals on 𝑝-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129 (1989), no. 2, 237–251. MR 986793, 10.2307/1971447
  • [4] Harish-Chandra, The Plancherel formula for reductive $ p$-adic groups (and corrections), Collected Papers, Vol. 4, pp. 353-370, preprint.
  • [5] R. Howe and A. Moy, Harish-Chandra homomorphisms for $ p$-adic groups, CBMS Regional Conf. Ser. in Math., vol. 59, Amer. Math. Soc., Providence, R.I., 1985.
  • [6] -, Minimal $ K$-types for $ G{l_n}$ over $ p$-adic field, Orbites Unipotentes et Représentations, Vol. II: Groupes $ p$-Adiques et Réels, Astérique, no. 171-172, 1989.
  • [7] D. Jabon, The supercuspidal representations of $ U(2,1)$ and $ GS{p_4}$ over a local field via Hecke algebra isomorphisms, Thesis, Univ. of Chicago, 1989.
  • [8] David Keys, Principal series representations of special unitary groups over local fields, Compositio Math. 51 (1984), no. 1, 115–130. MR 734788
  • [9] C. David Keys and Freydoon Shahidi, Artin 𝐿-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 67–89. MR 944102
  • [10] 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
  • [11] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
  • [12] I. G. Macdonald, Spherical functions on a group of 𝑝-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. MR 0435301
  • [13] Lawrence Morris, Tamely ramified supercuspidal representations of classical groups. I. Filtrations, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 705–738. MR 1142907
  • [14] Allen Moy, Representations of 𝑈(2,1) over a 𝑝-adic field, J. Reine Angew. Math. 372 (1986), 178–208. MR 863523, 10.1515/crll.1986.372.178
  • [15] Gopal Prasad and M. S. Raghunathan, Topological central extensions of semisimple groups over local fields, Ann. of Math. (2) 119 (1984), no. 1, 143–201. MR 736564, 10.2307/2006967
  • [16] 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, 10.2307/1971524
  • [17] 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
  • [18] Allan J. Silberger, Special representations of reductive 𝑝-adic groups are not integrable, Ann. of Math. (2) 111 (1980), no. 3, 571–587. MR 577138, 10.2307/1971110
  • [19] Allan J. Silberger, Complementary series for 𝑝-adic groups. I, Trans. Amer. Math. Soc. 259 (1980), no. 2, 589–598. MR 567099, 10.1090/S0002-9947-1980-0567099-5

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DOI: https://doi.org/10.1090/S0002-9947-1994-1106191-3
Article copyright: © Copyright 1994 American Mathematical Society