A computation about Whittaker functions and Eisenstein series 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.
 A: I am not entirely sure what you are asking for question 2, but let me take a crack at (1). At verious points I will be sloppy about distiguishing between a coset and its chosen representative, but it shouldn't cause any confusion.
(1) We have $B(F)\backslash G(F) = \cup_{w\in W}B\backslash BwU$. To get the coset reps, we need to see when we can have $$bwu=b'w'u'\quad \text{or, more simply} \quad wu=bw'.$$
Since we know that the Bruhat decomposition is a disjoint union, for this to occur we must have $w'=w$, reducing the identity to $wuw^{-1}=b$. Note that this forces $b\in U(F)$, since $b\in B(F)$ and $wuw^{-1}\in G(F)$ is unipotent.
Thus, the elements of $U(F)$ such that $B(F)wu = B(F)w$ is precisely $(U(F)\cap wU(F)w^{-1})$. Hence, to get coset representatives for the Bruhat cell corresponding to $w\in W$, we need to select representatives of the quotient $$(U(F)\cap wU(F)w^{-1})\backslash U(F).$$
For (2), it seems that all they are doing is the standard change of variables $u\mapsto u(u')^{-1}$ with $u$ a representative in the quotient $$(U(A)\cap wU(A)w^{-1})\backslash U(A),$$ and $u'$ a representative of the quotient $$(U(F)\cap wU(F)w^{-1})\backslash (U(A)\cap wU(A)w^{-1}).$$
The reason $u'$ does not appear in the argument for $f$ is that $u'\in (U(A)\cap wU(A)w^{-1})$ allows you to push it past $w^{-1}$ and use the invariance properties of $f$ as an element of the parabolocally induced representation to get rid of it.
