Decay of matrix coefficients of non-tempered representation

Question

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!

Answer 1

There is some confusion here, as literally the construction of complementary series in $SL_{2}$ will give you unitary representations with arbitrary slow decay. For any homogeneous space $G/\Gamma$, there exists a ``spectral gap'' in your definition. The statement about Selberg's work is about showing a uniform spectral gap for a tower of coverings amounting to principal congruence subgroups, not just about the dual of $SL_{2}$ say.

Now for $n\geq 3$, $SL_{n},PGL_{n}$ are Kazhdan groups, hence such an estimate is immediate for all (non-trivial) unitary representations, without pertaining to a particular lattice/tower of coverings etc. You essentially can recover such a bound from the proof of Cowling-Howe-Haagrup (which implement Howe's technique to show temperedeness).

The exact statement appears in Hee Oh's famous paper about decay estimates of matrix coefficients - https://gauss.math.yale.edu/~ho2/doc/mat.pdf as Theorem 2.5 (she defines the theorem for $K$-finite vectors which are usually enough, with some extra work in light of HC results you can probably recover something for smooth vectors, if you will use some Sobolev norms). See also Shalom's annals paper for related results.