A notion of rank for unitary representations of general linear groups
Author:
Roberto Scaramuzzi
Journal:
Trans. Amer. Math. Soc. 319 (1990), 349379
MSC:
Primary 22E50; Secondary 22E46
MathSciNet review:
958900
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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.
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 , Small unitary representations of classical groups, preprint.
 [H4]
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 R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), 7296. MR 533220 (80g:22017)
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 D. Kazhdan, Connection of dual space of a group with the structure of its closed subgroups, Functional Anal. Appl. 1 (1967), 6365. MR 0209390 (35:288)
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 G. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265301. MR 0098328 (20:4789)
 [M2]
 , Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101139. MR 0044536 (13:434a)
 [R]
 M. Rieffel, Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner, Studies in Analysis (G. C. Rota, ed.), Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York, pp. 4382. MR 546802 (81h:22004)
 [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]
 A. Silberger, Introduction to harmonic analysis on reductive adic groups, Princeton Univ. Press, Princeton, N.J., 1979. MR 544991 (81m:22025)
 [St]
 E. Stein, Analysis in matrix space and some new representations of , Ann. of Math. (2) 86 (1967), 461490. MR 0219670 (36:2749)
 [T]
 M. Tadić, Solution of the uniterizability problem for general linear group (nonArchimedean case), preprint.
 [V]
 D. Vogan, The unitary dual of over an Archimedean field, preprint. MR 827363 (87i:22042)
 [VN]
 J. von Neumann, Die Eindeutgkeit der Schràderschen Operatores, Math. Ann. 104 (1931), 570578.
 [Wg]
 S. P. Wang, The dual space of semisimple Lie groups, Amer. J. Math. 91 (1969), 921937. MR 0259023 (41:3665)
 [W]
 G. Warner, Harmonic analysis on semisimple Lie groups, Springer, 1972.
 [Z]
 G. J. Zuckerman, Continuous cohomology and unitary representations of real reductive groups, Ann. of Math. (2) 107 (1978), 495516. MR 496844 (81c:22025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009589008
PII:
S 00029947(1990)09589008
Article copyright:
© Copyright 1990
American Mathematical Society
