Continuous measures on compact Lie groups
Annales de l'Institut Fourier, Volume 50 (2000) no. 4, p. 1277-1296

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 ν such that ν*μ is absolutely continuous with respect to the Haar measure on G for every μ in C. In Section 4 we show that if f is a finite linear combination of characters then there exist two singular measures μ and ν on G such that f=μ*ν. In the final section we obtain a Wiener-type characterization of a continuous measure on a symmetric space of compact type G/K.

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 ν telle que la mesure ν*μ soit absolument continue par rapport à la mesure de Haar sur G pour toute mesure μ dans C. À la section 4 on montre que si f est une combinaison linéaire finie de caractères, il existe deux mesures singulières μ et ν sur G telles que f=μ*ν. À la section 5 on donne une caractérisation des mesures continues sur un espace symétrique compact G/K.

@article{AIF_2000__50_4_1277_0,
     author = {Anoussis, M. and Bisbas, A.},
     title = {Continuous measures on compact Lie groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     pages = {1277-1296},
     doi = {10.5802/aif.1793},
     zbl = {0969.43001},
     mrnumber = {2001m:43020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_4_1277_0}
}
Anoussis, M.; Bisbas, A. Continuous measures on compact Lie groups. Annales de l'Institut Fourier, Volume 50 (2000) no. 4, pp. 1277-1296. doi : 10.5802/aif.1793. http://www.numdam.org/item/AIF_2000__50_4_1277_0/

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