# Decay of matrix coefficients of non-tempered representation

A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $$\pi$$ of a semi-simple algebraic $$G$$ in terms of Harish-Chandra $$\Xi$$ function, of the form $$\langle \pi(g) u,v\rangle \ll_{u,v} \Xi(g),\quad u,v\in \pi\setminus\pi^G, g\in G.$$ Does there exist a similar, possibly weaker, bound in case of non-tempered representations (possibly of the above form but $$\Xi(g)$$ is replaced by $$\Xi(g)^{1-\delta}$$ for some $$0<\delta<1$$)?

I am, in particular, interested in $$\mathrm{GL}(n)$$ and $$\pi$$ being an irreducible automorphic representation of it. For $$n=2$$ it is known that such $$\delta$$ exists (spectral gap) due to the works of Selberg, Gelbart--Jacquet, Kim--Shahidi.