Harmonic analysis of spherical functions on $SU(1,1)$

Benyamini, Y.; Weit, Yitzhak

Annales de l'Institut Fourier, Tome 42 (1992) no. 3, pp. 671-694.

Soit $L^1(K\setminus G/K)$ l’algèbre des fonctions sphériques intégrales sur $SU(1,1)$, munie de l’opération de convolution comme multiplication. C’est une algèbre commutative semi-simple. Nous utilisons la transformation de Gelfand pour étudier les idéaux de $L^1(K\setminus G/K)$. En particulier, nous trouvons des conditions sur un idéal qui garantissent qu’il est identique à $L^1(K\setminus G/K)$, ou à $L_0^1(K\setminus G/K)$.

Les fonctions sphériques sur $SU(1,1)$ se représentent naturellement comme des fonctions radiales sur le disque unité $D$ du plan complexe. À l’aide de cette représentation, nous appliquons les résultats précédents à la caractérisation des fonctions harmoniques et holomorphes sur $D$.

Denote by $L^1(K\setminus G/K)$ the algebra of spherical integrable functions on $SU(1,1)$, with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in $L^1(K\setminus G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^1(K\setminus G/K)$, or that it is $L_0^1(K\setminus G/K)$. Spherical functions on $SU(1,1)$ are naturally represented as radial functions on the unit disk $D$ in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on $D$.

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