Tempered Subgroups and Representations with Minimal Decay of Matrix Coefficients

We present a function F for each simple real linear Lie group G with real rank at least 2 such that F bounds from above all the K-matrix coeecients of non-trivial irreducible spherical unitary representations of G where K is a maximal compact subgroup of G. This enables us to determine when a closed subgroup H is a (G; K)-tempered subgroup of G: for example, if the restriction Fj H of F to H lies in L 1? (H). We also prove that this function F is the best possible for G a real-split group of type A n or C n and as a consequence, we obtain that if H is semisimple, then H is a (G; K)-tempered subgroup of G if and only if Fj H lies in L 1 (H).