A theorem of [Cowling--Haagerup--Howe][1] 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.

Thanks in advance!


  [1]: http://www.math.univ-metz.fr/~gnc/bibliographie/HarmonicAnalysis/Cowling-1988-Almost%20L2%20matrix%20coefficients.pdf