Continuous measures on compact Lie groups

Anoussis, M.; Bisbas, A.

Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1277-1296.

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 $\nu *\mu$ 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 $f=\mu *\nu$. À la section 5 on donne une caractérisation des mesures continues sur un espace symétrique compact $G/K$.

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 $\nu *\mu$ 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 $f=\mu *\nu$. In the final section we obtain a Wiener-type characterization of a continuous measure on a symmetric space of compact type $G/K$.

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