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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 [HM]
 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)
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 , Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101139. MR 0044536 (13:434a)
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 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
