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Journal of Lie Theory, Vol. 10, No. 2, pp. 255-267 (2000)
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A proof of property (RD) for cocompact lattices of SL(3,R) and SL(3,C)

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Vincent Lafforgue

Département de Mathématiques et Applications

Ecole Normale Supérieure

45 rue d'Ulm

75230 PARIS CEDEX 05, France

vlafforg@dmi.ens.fr

**Abstract:** We prove that cocompact lattices in $SL_3(\mathbb R)$ and $SL_3(\mathbb C)$ have property (RD) of Jolissaint. This property asserts that functions on these groups which are $l^2$ with some polynomial decay belong to the reduced $C^*$-algebra. Ramagge, Robertson and Steger proved the same result for cocompact lattices in $SL_3$ of $p$-adic fields and we use the same method.

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