I'm working through chapter IV of Macdonald's book, finding it somewhat complex due to its use of notations and concepts from earlier chapters. To get a clearer understanding, I looked up a review by Stanley, available on his website: https://math.mit.edu/~rstan/pubs/pubfiles/49.pdf.
In page 260, he writes: Let $\lambda=(\lambda_1,\dots,\lambda_l)$ be a partition of $n$ and let $P=P_\lambda$ be the parabolic subgroup of $G=\mathrm{GL}_n(\mathbb{F}_q)$ associated to $\lambda$. Let $\eta_\lambda(q)$ be the character of $\operatorname{Ind}_{P}^G(1)$. He claims there exist irreducible characters $\chi^\mu(q)$ of $\operatorname{GL}_n(\mathbb{F}_q)$ such that
$$\eta_\lambda(q)=\sum_{\mu}K_{\lambda\mu}\chi^\mu(q). \tag{1}\label{476211_1}$$
He also states that the value of $\chi^\lambda(q)$ at a unipotent element of type $\mu$ is$$q^{n(\mu)-n(\lambda)}K_{\lambda\mu}(q^{-1}).\tag{2}\label{476211_2}$$
I have two questions about these claims:
Regarding formula \eqref{476211_1}, if we take $\lambda = (1,1,\dots,1)$, meaning $P_\lambda$ is the Borel subgroup $B$, then $K_{\lambda\mu} = 0$ unless $\mu = \lambda$, where $K_{\lambda\lambda} = 1$. This suggests that $\operatorname{Ind}_{B}^G(1)$ is irreducible which I am not sure if this is correct. Could there be a typo in Stanley's claim, or am I overlooking something? What would be the accurate statement if the claim needs correction?
Regarding formula \eqref{476211_2}, I couldn't find this assertion in Macdonald's book. Where can I find a reference for this?