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Pointwise estimates of the size of characters of compact Lie groups

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Kathryn E. Hare ,
David C. Wilson  and
Wai Ling Yee
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics University of WaterlooWaterloo, OntCanada e-mail: [email protected]
David C. Wilson
Affiliation:
Po Box 280 Churchill VIC 3842Australia e-mail: [email protected]
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Abstract

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Pointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained with which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L2. We also prove that if k = rank G then μ 2k ∈ L1 for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be dimension of the group G.

Keywords

central measurescompact Lie groupcharacters

MSC classification

Secondary: 43A80: Analysis on other specific Lie groups 22E46: Semisimple Lie groups and their representations 43A65: Representations of groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 1 , August 2000 , pp. 61 - 84
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Brocker, T. and Dieck, T., Representations of compact Lie groups (Springer, New York, 1985).Google Scholar
[2]Hare, K., ‘The size of characters of compact Lie groups’, Studia Math. 129 (1998), 1–18.Google Scholar
[3]Hare, K., ‘Central Sidonicity for compact Lie groups’, Ann. Inst. Fourier (Grenoble) 45 (1995), 547–564.CrossRefGoogle Scholar
[4]Hare, K., ‘Properties and examples of (Lp, Lq) mulitpliers’, Indiana Univ. Math. J. 338 (1989), 211–227.CrossRefGoogle Scholar
[5]Humphreys, J., Introduction to Lie algebras and representation theory (Springer, New York, 1972).Google Scholar
[6]Mckay, W., Patera, J. and Rand, D., Tables of representation of simple Lie algebras, vol. 1 Exceptional simple Lie algebras, Centre de Recherches Math., Univ. of Montreal, 1990.Google Scholar
[7]Mimura, M. and Toda, H., Topology of Lie groups, Transl. Math. Monographs 91 (Amer. Math. Soc., Providence, R.I,, 1991).Google Scholar
[8]Ragozin, D., ‘Central measures on compact simple Lie groups’, J. Funct. Anal. 10 (1972), 212–229.Google Scholar
[9]Ricci, F. and Stein, E. M., ‘Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds’, J. Funct. Anal. 86 (1989), 360–389.Google Scholar
[10]Ricci, F. and Travaglini, G., ‘Lp − Lq estimates for orbital measures and Radon transform on compact Lie groups and Lie algebras’, J. Funct. Anal. 129 (1995), 132–147.Google Scholar
[11]Rider, D., ‘Central lacunary sets’, Monatsh. Math. 76 (1972), 328–338.Google Scholar
[12]Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Eculidean spaces (Princeton Univ. Press, Princeton, 1971).Google Scholar
[13]Varadarajan, V., Lie groups, Lie algebras and their representations (Springer, New York, 1984).Google Scholar