Let $G$ be a split reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of G. Let $\chi$ be an unramified character of $M$ and $f_\chi$ be the spherical section of the normalized induced representation $I^G_P(\chi)$. I'm interested in explicitly writing down the constant term of the spherical Eisenstein series $$ E(f_\chi)=\sum_{\gamma\in P(F)\backslash G(F)} f_\chi(\gamma g) $$ along a standard parabolic subgroup $Q=UL$. Of course, this involves doing the decomposition $P\backslash G/Q=\sqcup_{w} PwQ$ and computing adelic integrals of the type $$ f_{\chi,w}(g):=\int_{U_w\backslash U}f_{\chi}(wug)du,\quad (*) $$ where $U_w=w^{-1}Pw\cap U$. The function $f_{\chi,w}(g)$ is spherical and falls in certain induced representation; the issue is to evaluate $f_{\chi,w}$ at $1$.
When $Q$ is the Borel subgroup, the integral $(*)$ is computed by the Gindikin-Karpelevich formula. For example, the local p-adic factor of $f_{\chi,w}(1)$ is given by $$ \prod_{\alpha>0, U_\alpha\subset U-U_w} \frac{1-p^{-1}\chi(H_\alpha)}{1-\chi(H_\alpha)}, $$ where $\alpha$ denotes a root, $U_\alpha$ is the associated unipotent subgroup, and $H_\alpha$ is the coroot.
My questions are:
Is there a similar easy-to-use formula when Q is a general parabolic subgroup?
Is there any reference on determining the poles of the constant term of $E(f_\chi)$ along $\mathrm Q$, say in the situation split symplectic or orthogonal group?
