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Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete data, for example, a $\mathrm{GL}(5)$ Eisenstein series which is induced from $2+2+1$ parabolic with data $\{\text{cusp form}, \text{residual}, |.|^s\}$.

I know that Langlands proved Maass--Selberg for any generic Eisenstein series, i.e., induced from cuspidal data, but I am not sure if that proof includes/can be extended to non-generic Eisenstein series, as well. The only example I know about Maass--Selberg relation with non-generic Eisenstein series is for $n=3$ which is in S. D. Miller's thesis.

For motivation, one may think about the question: how to estimate $\|\varphi\|_{L^2(\Omega)}$ where $\Omega$ is a fixed compact set in the fundamental domain and $\varphi$ is any standard automorphic form i.e. appears in the $L^2$ spectral decomposition for $\mathrm{GL}(n)$. I believe that this quantity would be of size $\ll_\epsilon C(\varphi)^\epsilon$ where $C(\varphi)$ is the conductor of $\varphi$.

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    $\begingroup$ This question is somewhat answered in a paper of Arthur: doi.org/10.1215/S0012-7094-82-04904-3. There is a Maass-Selberg identity stated at the beginning of Section 9; in general, this can be quite complicated, and Arthur's main goal is just to understand the asymptotics of this as the truncation parameter $T$ tends to infinity. But I imagine for $G = \mathrm{GL}_n$, one could actually make this reasonably explicit. $\endgroup$
    – Peter Humphries
    Commented Oct 13, 2020 at 2:06

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