Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A notion of rank for unitary representations of general linear groups

Author: Roberto Scaramuzzi
Journal: Trans. Amer. Math. Soc. 319 (1990), 349-379
MSC: Primary 22E50; Secondary 22E46
MathSciNet review: 958900
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.

References [Enhancements On Off] (What's this?)

  • [B] Joseph N. Bernstein, 𝑃-invariant distributions on 𝐺𝐿(𝑁) and the classification of unitary representations of 𝐺𝐿(𝑁) (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50–102. MR 748505, 10.1007/BFb0073145
  • [BZ] I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive 𝔭-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 0579172
  • [F1] J. M. G. Fell, The dual spaces of 𝐶*-algebras, Trans. Amer. Math. Soc. 94 (1960), 365–403. MR 0146681, 10.1090/S0002-9947-1960-0146681-0
  • [F2] J. M. G. Fell, Weak containment and induced representations of groups, Canad. J. Math. 14 (1962), 237–268. MR 0150241
  • [HC] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 0219666
  • [H1] Roger Howe, On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223–331. MR 777342
  • [H2] R. Howe, 𝜃-series and invariant theory, 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. 275–285. MR 546602
  • [H3] -, Small unitary representations of classical groups, preprint.
  • [H4] Roger Howe, The Fourier transform and germs of characters (case of 𝐺𝑙_{𝑛} over a 𝑝-adic field), Math. Ann. 208 (1974), 305–322. MR 0342645
  • [HM] Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), no. 1, 72–96. MR 533220, 10.1016/0022-1236(79)90078-8
  • [K] 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
  • [M1] George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 0098328
  • [M2] George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 0044536
  • [R] Marc A. Rieffel, Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner, Studies in analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pp. 43–82. MR 546802
  • [Sa1] S. Sahi, Spherical unitary representations of general linear groups over local fields, doctoral dissertation, Yale University, 1985.
  • [Sa2] -, On Kirillov's conjecture for Archimedean fields, preprint.
  • [S] R. Scaramuzzi, Unitary representations of small rank of general linear groups, doctoral dissertation, Yale University, 1985.
  • [Si] 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
  • [St] E. M. Stein, Analysis in matrix spaces and some new representations of 𝑆𝐿(𝑁,𝐶), Ann. of Math. (2) 86 (1967), 461–490. MR 0219670
  • [T] M. Tadić, Solution of the uniterizability problem for general linear group (non-Archimedean case), preprint.
  • [V] David A. Vogan Jr., The unitary dual of 𝐺𝐿(𝑛) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. MR 827363, 10.1007/BF01394418
  • [VN] J. von Neumann, Die Eindeutgkeit der Schràderschen Operatores, Math. Ann. 104 (1931), 570-578.
  • [Wg] S. P. Wang, The dual space of semi-simple Lie groups, Amer. J. Math. 91 (1969), 921–937. MR 0259023
  • [W] G. Warner, Harmonic analysis on semi-simple Lie groups, Springer, 1972.
  • [Z] Gregg J. Zuckerman, Continuous cohomology and unitary representations of real reductive groups, Ann. of Math. (2) 107 (1978), no. 3, 495–516. MR 496844, 10.2307/1971126

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E50, 22E46

Retrieve articles in all journals with MSC: 22E50, 22E46

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society