Valette's conjecture on property RD for groups acting on symmetric spaces or affine buildings

Let $\Gamma$ be a discrete group acting isometrically, properly and cocompactly either on a Riemannian symmetric space or on an affine building. The group has property RD if there exist $R>0$ and a polynomial $P$ such that, for every non-negative integer $n$ and every finitely supported complex-valued function $f$ on

$$S^{\Gamma}_{n,R}=\{\gamma\in\Gamma\mid nR\leq |\gamma|<(n+1)R\},$$

we have

$$\|\lambda_{\Gamma}(f)\|_{op}\leq P(n)\|f\|_{2}.$$

Valette's conjecture. Property RD holds for any discrete group acting isometrically, properly and cocompactly either on a Riemannian symmetric space or on an affine building. Property RD is known for many important classes of groups, including hyperbolic groups and cocompact lattices in rank-one real semisimple Lie groups, but the conjecture remains open in general.