Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)

I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it and I would appreciate any explanation.

On page 429 they define $$c(w, \lambda) = \int_{ (w' N_0(\mathbb{A})(w')^{-1} \cap N_0 (\mathbb{A})) \backslash N_0(\mathbb{A}) } exp() dn.$$ Then they say $$\int_{N_0(\mathbb{Q}) \backslash N_0(\mathbb{A})} E^G_{P_0}(\lambda, ng) dn = \sum_{w \in W} c(w, \lambda) exp( ) dn.$$ I think that this statement should follow from the Langlangds' result on page 11 in these slides by Shahidi with $$P = P' = P_0$$:
https://www.ima.umn.edu/materials/2018-2019/SW11.14-16.18/27644/Shahidi_Abhi.pdf (or another reference for this is Theorem 6.2.2. (p114) in Shahidi's book "Eisenstein series and autompic L-functions"). But I seem to be missing something and have not been able to deduce the above statement. For example, in the above $$c(w, \lambda)$$ is independent of $$g$$ while in the formula in the link it seems to depend on it... Any explanation would be appreciated. Thank you.

Some notation: $$P_0$$ is a minimal $$\mathbb{Q}$$ parabolic subgroup of $$G$$ semisimple linear algebraic group over $$\mathbb{Q}$$. $$P_0 = M_0 N_0$$ where $$M_0$$ is a Levi subgroup of $$P_0$$. And $$E^G_{P_0}$$ is the Eisenstein series $$E^G_{P_0}(\lambda, g) = \sum_{\gamma \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) } exp(<\lambda + \rho_{P_0}, H_{P_0} (\gamma g) >).$$

PS I am starting to wonder maybe $$c(w,\lambda)$$ in the paper (as above) is actually a typo(?) and that it should be replaced by $$M(w, \lambda)(1)(g) = \int_{ (w' N_0(\mathbb{A})(w')^{-1} \cap N_0 (\mathbb{A})) \backslash N_0(\mathbb{A}) } exp() exp() dn$$ as in the formula given in the link above. I am new to the subject and I'm not too sure as maybe I'm missing stuff. Any comments would be appreciated. Thank you.

I haven't carefully gone through your notations, but usually the constant term of Eisenstein series can be expressed as a sum, going over $$w$$ in some subset of Weyl group, of values of sections after applying appropriate intertwining operators. The last formula of $$M(w,\lambda)$$ seems about right to me.