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<blockquote data-quote="MO B" data-source="post: 740119">I&#039;m working through chapter IV of Macdonald&#039;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: <a href="https://math.mit.edu/%7Erstan/pubs/pubfiles/49.pdf">https://math.mit.edu/~rstan/pubs/pubfiles/49.pdf</a>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 $\mathrm{Ind}_{P}^G(1)$. He claims there exist irreducible characters $\chi^\mu(q)$ of $\mathrm{GL}_n(\mathbb{F}_q)$ such that$$ \eta_\lambda(q)=\sum_{\mu}K_{\lambda\mu}\chi^\mu(q) \qquad\qquad\qquad (1) $$He also states that the value of $\chi^\lambda(q)$ at a unipotent element of type $\mu$ is $$ q^{n(\mu)-n(\lambda)}K_{\lambda\mu}(q^{-1})\qquad\qquad\qquad\qquad (2) $$I have two questions about these claims:Regarding formula (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 $\mathrm{Ind}_{B}^G(1)$ is irreducible which I am not sure if this is correct. Could there be a typo in Stanley&#039;s claim, or am I overlooking something? What would be the accurate statement if the claim needs correction?Regarding formula (2), I couldn&#039;t find this assertion in Macdonald&#039;s book. Were can I find a reference for this?Many thanks for your help.</blockquote>

[QUOTE="MO B, post: 740119"] 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: [URL='https://math.mit.edu/%7Erstan/pubs/pubfiles/49.pdf']https://math.mit.edu/~rstan/pubs/pubfiles/49.pdf[/URL] 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 $\mathrm{Ind}_{P}^G(1)$. He claims there exist irreducible characters $\chi^\mu(q)$ of $\mathrm{GL}_n(\mathbb{F}_q)$ such that $$ \eta_\lambda(q)=\sum_{\mu}K_{\lambda\mu}\chi^\mu(q) \qquad\qquad\qquad (1) $$ He also states that the value of $\chi^\lambda(q)$ at a unipotent element of type $\mu$ is $$ q^{n(\mu)-n(\lambda)}K_{\lambda\mu}(q^{-1})\qquad\qquad\qquad\qquad (2) $$ I have two questions about these claims: Regarding formula (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 $\mathrm{Ind}_{B}^G(1)$ 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 (2), I couldn't find this assertion in Macdonald's book. Were can I find a reference for this? Many thanks for your help. [/QUOTE]

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