The principal series module for finite GL_n

Let $n\ge 1$, let $q$ be a prime power, put $G={\mathrm{GL}}_n({\mathbb{F}}_q)$ with standard Borel $B=T⋉U$ and diagonal torus $T$, and let $\chi \in \widehat{T}$ be a character of $T$ (Standard subgroups of finite general linear groups, Diagonal torus characters and the Weyl action). Since $B=T⋉U$ and $T\cong B\mathrm{/}U$ (Block Levi decomposition of standard parabolics, Compositions, partial flags, and standard parabolics), the projection $\pi :B\to T$, $\pi (lu)=l$, is a surjective homomorphism with kernel $U$. The inflation of $\chi$ from $T$ to $B$ is the character $\tilde{\chi }:=\chi \circ \pi :B⟶{\mathbb{C}}^{\times },\tilde{\chi }(lu)=\chi (l),$ which is trivial on $U$ (Harish-Chandra induction and restriction for finite general linear groups).

The principal series module attached to $\chi$ is the complex $G$-module $I(\chi ):=R_T^G(\chi )={\mathrm{Ind}}_B^G\left({\mathrm{Inf}}_T^B\chi \right)=\{f:G\to \mathbb{C}:f(gb)=\tilde{\chi }(b{)}^{-1}f(g)\mathrm{\forall }g\in G,b\in B\},$ with addition and scalar multiplication pointwise and $(g_0\cdot f)(g):=f(g_0^{-1}g)$ (The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$); here $R_T^G$ is Harish-Chandra induction from the split Levi $T$ with respect to $B$ (Harish-Chandra induction and restriction for finite general linear groups). Equivalently $I(\chi )={\mathrm{Ind}}_B^G(\tilde{\chi })$, the induction of the one-dimensional $B$-module ${\mathbb{C}}_{\tilde{\chi }}$.

Dimension. Since ${\dim }_{\mathbb{C}}{\mathbb{C}}_{\tilde{\chi }}=1$, the dimension formula for induced representations gives ${\dim }_{\mathbb{C}}I(\chi )=[G:B]$ (The dimension of an induced finite-dimensional representation is $[G:H]\dim W$). The index $[G:B]$ is the number of complete flags of ${\mathbb{F}}_q^n$ (Complete flags are G/B), and it equals ${\prod }_{i=1}^n\frac{q^i-1}{q-1}$: choosing the columns of a matrix in $G$ successively gives $\mathrm{\mid }G\mathrm{\mid }=(q^n-1)(q^n-q)\cdots (q^n-q^{n-1})=q^{n(n-1)\mathrm{/}2}{\prod }_{i=1}^n(q^i-1)$, while $\mathrm{\mid }B\mathrm{\mid }=(q-1{)}^n{q}^{n(n-1)\mathrm{/}2}$ as recorded in Standard subgroups of finite general linear groups, and the quotient is the displayed product. In particular $I(1)=R_T^G(1)$ for the trivial character $1$ of $T$.

Dependence on $\chi$. The module $I(\chi )$ depends on the chosen representative $\chi$ of its $S_n$-orbit, and the relation between the modules $I(\chi )$ and $I(w\cdot \chi )$ for $w\in S_n$ is examined together with the endomorphism algebra of $I(\chi )$ in the results below; the definition itself fixes one character and one module.