Poles and residues of degenerate Eisenstein series on GL(n)

Question

Suppose $P \subseteq GL(n)$ is a parabolic subgroup. How do I find (or what is a reference for) the poles and residues of degenerate Eisenstein series associated to $P$?

More precisely, suppose $P_1, \ldots, P_r$ are parabolic subgroups of $GL(n)$ containing $P$, $\delta_{P_i}$ denotes the modulus character of $P_i$, $s_1, \ldots, s_r$ are complex parameters, and $$f(g, s_1, \ldots, s_r) \in Ind_{P}^{GL(n)}(\delta_{P_1}^{s_1} \cdots \delta_{P_r}^{s_r}).$$ The degenerate Eisenstein series associated to this data is $$E(g,f,s_1, \ldots, s_r) = \sum_{\gamma \in P(F)\backslash GL_n(F)}{f(\gamma g, s_1, \ldots, s_r)}.$$

Question. Where are the poles of this Eisenstein series located, and what are the residues at these poles?

If $P$ is the maximal parabolic of $GL(n)$ that stabilizes a line in the $n$-dimensional representation of $GL(n)$, the above question may be answered using Poisson summation. Outside of this special case, any comments or references would be helpful.

Answer 1

First, because the kind of induction that creates Eisenstein series can be done in stages, and commutes (after meromorphic continuation) with taking residues (even in the generalized sense required in this story), all these degenerate Eisenstein series are residues of the minimal-parabolic Eisenstein series.

Second, the poles (or polar divisors) of Eisenstein series (of all sorts) only occur at poles of their constant terms. The minimal-parabolic Eisenstein series has a very symmetrical minimal-parabolic constant term (and all the other constant terms are related by another sort of induction).

(Thus,) examination of that minimal-parabolic constant term of the minimal-parabolic Eisenstein series shows the hyperplanes along which that Eisenstein series, and any of its residues, can have poles.

An appendix in Langlands 544 treats that minimal parabolic Eis via Poisson summation and Bochner's analytic continuation lemma. Essentially the same argument is given in http://www.math.umn.edu/~garrett/m/v/gln_cont_spec.pdf

Then, for subtler reasons (basically looking at Plancherel), I think none of the residues of these Eis except the constant function are square-integrable. Going farther out "on a limb", it may be that the only residues in the cone analogous to a right half-plane are even-more-degenerate Eisenstein series. Poles outside that cone are more complicated, due to the functional equations.