Continuous measures on compact Lie groups

Anoussis, M. ; Bisbas, A.

Annales de l'Institut Fourier, Tome 50 (2000), p. 1277-1296 / Harvested from Numdam

Résumé

On étudie des mesures continues sur un groupe de Lie compact semi-simple $G$ en utilisant la théorie des représentations. À la section 2 nous donnons une caractérisation des mesures continues sur $G$ analogue à celle de Wiener pour le groupe du cercle. Puis on construit des mesures sur $G$ qui sont liées aux produits de Riesz sur les groupes abéliens localement compacts. En utilisant cette mesure on montre à la section 3 que si $C$ est une partie compacte des mesures continues sur $G$, il existe une mesure singulière $\nu $ telle que la mesure $\nu \ast \mu $ soit absolument continue par rapport à la mesure de Haar sur $G$ pour toute mesure $\mu $ dans $C$. À la section 4 on montre que si $f$ est une combinaison linéaire finie de caractères, il existe deux mesures singulières $\mu $ et $\nu $ sur $G$ telles que $f=\mu \ast \nu $. À la section 5 on donne une caractérisation des mesures continues sur un espace symétrique compact $G/K$.

Abstract

We study continuous measures on a compact semisimple Lie group $G$ using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on $G$ which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if $C$ is a compact set of continuous measures on $G$ there exists a singular measure $\nu $ such that $\nu \ast \mu $ is absolutely continuous with respect to the Haar measure on $G$ for every $\mu $ in $C$. In Section 4 we show that if $f$ is a finite linear combination of characters then there exist two singular measures $\mu $ and $\nu $ on $G$ such that $f=\mu \ast \nu $. In the final section we obtain a Wiener-type characterization of a continuous measure on a symmetric space of compact type $G/K$.

Détail

Publié le : 2000-01-01

MR 2001m:43020 | Zbl 0969.43001 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1793

Bibliographie

[1] A. Bisbas and C. Karanikas, On the continuity of measures, Applicable Analysis, 48 (1993), 23-35. | MR 95e:43001 | Zbl 0792.42014

[2] M. Blümlinger, Rajchman measures on compact groups, Math. Ann., 284 (1989), 55-62. | MR 90f:22003 | Zbl 0673.43005

[3] N. Bourbaki, Intégration, Hermann, Paris, 1963.

[4] T. Bröcker and T. Tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985. | Zbl 0581.22009

[5] A. H. Dooley and S. K. Gupta, Continuous singular measures with absolutely continuous convolution squares, Proc. Amer. Math. Soc., 124 (1996), 3115-3122. | MR 96m:43011 | Zbl 0861.43006

[6] C. F. Dunkl and D. E. Ramirez, Helson sets in compact and locally compact groups, Mich. Math. J., 19 (1971), 65-69. | MR 48 #2667 | Zbl 0225.43009

[7] J. M. G. Fell and R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Volume 2, Banach *-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis, Academic Press, London, 1988. | Zbl 0652.46051

[8] C. C. Graham and A. Maclean, A multiplier theorem for continuous measures, Studia Math., LXVII (1980), 213-225. | MR 81k:43006 | Zbl 0434.43006

[9] C. C. Graham and O. C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer-Verlag, New York, 1979. | MR 81d:43001 | Zbl 0439.43001

[10] K. E. Hare, The size of characters of compact Lie groups, Studia Math., 129 (1998), 1-18. | MR 99c:43013 | Zbl 0946.43006

[11] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. | MR 80k:53081 | Zbl 0451.53038

[12] S. Helgason, Groups and Geometric Analysis, Academic Press, London, 1984. | Zbl 0543.58001

[13] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag, Berlin, 1970. | Zbl 0213.40103

[14] E. Hewitt and K. Stromberg, A remark on Fourier-Stieltjes transforms, An. da Acad. Brasileira de Ciencias, 34 (1962), 175-180. | MR 27 #533 | Zbl 0117.09601

[15] V. Hösel and R. Lasser, A Wiener theorem for orthogonal polynomials, J. Funct. Anal., 133 (1995), 395-401. | MR 97b:42041 | Zbl 0843.42011

[16] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. | MR 48 #2197 | Zbl 0254.17004

[17] D. L. Ragozin, Central measures on compact simple Lie groups, J. Funct. Anal., 10 (1972), 212-229. | MR 49 #5715 | Zbl 0286.43002

[18] D. L. Ragozin, Zonal measure algebras on isotropy irreducible homogeneous spaces, J. Funct. Anal., 17 (1974), 355-376. | MR 51 #1297 | Zbl 0297.43002

[19] D. Rider, Central lacunary sets, Monatsh. Math., 76, (1972), 328-338. | MR 51 #3801 | Zbl 0258.43008

[20] R. S. Strichartz, Wavelet Expansions of Fractal Measures, The Journal of Geometric Analysis, 1 (1991), 269-289. | MR 93c:42036 | Zbl 0734.28011

[21] V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Springer-Verlag, New-York, 1984. | MR 85e:22001 | Zbl 0955.22500