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.

Thanks in advance!


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