I am reading The unramified principal series of $p$-adic groups by W. Casselman and am stuck on some basic details. $G$ is a connected, reductive group over a $p$-adic field, $P = MN$ is a minimal parabolic subgroup, $B$ is an Iwahori subgroup (not a Borel subgroup), and $\delta$ is the modulus character of $P$.
I am trying to understand the claim "$\phi_w$ is identically zero off $PwB$ and $\phi_w(pwb) = \chi \delta^{\frac{1}{2}}(p)$ for $p \in P, b \in B$."
Let $g \in G$, and first suppose $g \not\in PwB$. Then
$$\phi_{w,\chi}(g) = \int\limits_P \chi^{-1} \delta^{\frac{1}{2}}(p) \textrm{ch}_{BwB}(pg) \space dp$$
and I want to say that this integral is zero. Since $g \not\in PwB$, neither is $pg$ for any $p \in P$. Are we supposed to conclude here that $pg \not\in BwB$? This isn't clear to me.
Next, suppose $g \in PwB$, equal to $p_0wb$. Then
$$\phi_{w,\chi}(p_0wb) = \int\limits_P \chi^{-1} \delta^{\frac{1}{2}}(p) \textrm{ch}_{BwB}(pp_0wb) \space dp = \delta(p_0) \int\limits_P\chi^{-1} \delta^{\frac{1}{2}}(pp_0^{-1}) \textrm{ch}_{BwB}(pwb) \space dp$$
$$ = \chi \delta^{\frac{1}{2}}(p_0) \int\limits_P\chi^{-1} \delta^{\frac{1}{2}}(p) \textrm{ch}_{BwB}(pwb) \space dp$$
So we need to show that
$$\int\limits_P\chi^{-1} \delta^{\frac{1}{2}}(p) \textrm{ch}_{BwB}(pwb) \space dp = 1$$
which is also not clear to me.
Define $P_0 = P \cap K = P \cap B$, and similarly $$M_0 = M \cap K = M \cap B, \space \space \space \space \space \space N_0 = N_0 \cap K = N_0 \cap B$$ Also, let $N^-$ be the opposite unipotent radical of $N$, and define $N_1^- = N^- \cap B$ (this is not the same as $N^- \cap K$). The first claim follows from the Iwahori decomposition:
$$B = P_0N_1^- = N_1^-P_0$$
We want to show that if $g \not\in PwB$, then $pg \not\in BwB$ for any $p \in P$. Equivalently, we want to show:
Lemma 1: $BwB \subseteq PwB$.
Let $bwb' \in BwB$. Write $b = p_0n^-$ for some $p_0 \in P$ and $n^- \in N_1^-$. Then
$$bwb' = p_0n^-wb' = p_0w(w^{-1}n^-w)b'$$
which is in $PwB$, since $p_0 \in P$, and conjugation by $w^{-1}$ keeps $N_1^-$ inside $B$. I don't know how to prove this formally, but it is definitely true: imagine $N_1^-$ consists of all lower triangular unipotent matrices whose entries are integral with valuation $\geq 1$; conjugation by a permutation matrix will produce a matrix in $K$ whose lower left entries are either $0$ or are integral with value $\geq 1$. $\blacksquare$
This follows from Proposition 1.3 in the same paper and the lemma below. Proposition 1.3 says that
$$BwB \cap PwP = P_0wN_0$$
Lemma 2: Let $p \in P, b \in B$. Then $pwb \in BwB$ if and only if $p \in P_0$.
First, suppose that $pwb \in BwB$. Then so is $pw$, and hence $pw \in BwB \cap PwP = P_0wN_0$. We can then write $pw = p_0wn_0$ for $p_0 \in P_0, n_0 \in N_0$, and so $p = p_0wn_0w^{-1} \in P \cap K = P_0$.
Conversely, assume that $p \in P_0$. We are done, because $P_0 \subseteq B$. $\blacksquare$
Now
$$\int\limits_P\chi^{-1} \delta^{\frac{1}{2}}(p) \textrm{ch}_{BwB}(pwb) \space dp = \int\limits_{P_0}\chi^{-1} \delta^{\frac{1}{2}}(p) dp = \int\limits_{P_0} dp =1 $$
where we have used the fact that $\chi$ and $\delta$ are both trivial on $P_0 = P \cap K$ ($\chi$ is assumed to be unramified, and the modulus character is always trivial on compact subgroups), and the Haar measure on $P$ is normalized so that $P_0$ has measure $1$.