I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3.

My questions are in the following.

(1) I think that $B(F) \backslash G(F) = \cup_{w \in W} B \backslash BwB = \cup_{w \in W} B \backslash BwU = \cup_{w \in W} \{B wu : u \in U\}$. But why representatives for $B(F) \backslash G(F)$ map be taken to be the set of $w^{-1}v$ where $w \in W$ and for each $w$, $v$ runs through a set of representatives for $$(U(F)\cap wU(F)w^{-1})\backslash U(F)?$$ Why here $v$ runs through a set of representatives for $(U(F)\cap wU(F)w^{-1})\backslash U(F)$ but not $U(F)$?

(2) Why we have $$ \sum_{w \in W} \int_{(U(F) \cap w U(F) w^{-1})\backslash U(A)} f_{\zeta}(w^{-1} u g) \psi(u)^{-1} du = \\ \sum_{w \in W} \int_{(U(A) \cap wU(A)w^{-1})\backslash U(A)} \int_{(U(F) \cap wU(F)w^{-1})\backslash (U(A) \cap wU(A)w^{-1})} \psi(u')du' f_{\zeta}(w^{-1}ug) \psi(u)^{-1} du? $$

Thank you very much.