Region of convergence of Eisenstein series is a union of Weyl chambers when groups have discrete series?

Question

Let $G$ be a reductive algebraic group, and let $M$ be a Levi subgroup of $G$. In Urban's paper Eigenvarieties for Reductive Groups, it seems to be assumed that if $G(\mathbb{R})$ and $M(\mathbb{R})$ both have discrete series, then the region of convergence of Eisenstein series for the pair ($G$,$M$) is (up to translation) a union of Weyl chambers. (Otherwise it would not make sense to define the set $W^M_{Eis}$.) This assumption seems to be true in all cases that I have checked but I am not sure how to prove it except by exhaustive case analysis. Is a proof written down anywhere?

EDIT: I discussed this with Urban and it seems that the assumption is not true in general. For example, it is not true when $G=Sp(6)$ or $SO(4,3)$ and $M=GL(2) \times GL(1) \subset GL(3)$. However, the assumption does seem to be true when $M$ is maximal (and we still require $G$ and $M$ to have discrete series). I would still be interested in seeing a proof in that case.

Answer 1

The assumption in Urban's paper is incorrect: the region of convergence need not be a union of Weyl chambers. Hence the character distribution of the Eistenstein series coming from a single cusp generally does not have a unique $p$-adic interpolation. However, the sum of all of the character distributions will still have a unique $p$-adic interpolation. This can be seen as follows. The sum in Lemma 4.6.2 groups together the parabolic subgroups that are Weyl conjugates. If one instead groups together the parabolic subgroups that contain the same Levi, then the portions of the regions of convergence that satisfy $w=w_0$ fit together to form exactly one Weyl chamber.

In the case where M is maximal and G and M have discrete series, Corollary 5.6 in On the Eisenstein Series of Cohomology of Arithmetic Groups by Li and Schwermer and its Poincaré dual imply that the region of convergence is the union of translates of the fundamental Weyl chamber by elements of $W^M$ whose length is less than half of the largest length. Here $W^M$ is the set of minimal length coset representatives of $W_G/W_M$ and $W_G$ and $W_M$ are the Weyl groups of $G$ and $M$ respectively.