Quasi-regular representations and property RD

We study \emph{property RD} in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a $L^{1}$-integra\-bility condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of quasi-regular representations which satisfy a "non-RD condition" for non-uniform lattices. We obtain also that such coefficients can not satisfy \emph{the weak inequality} of Harish-Chandra.

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