Regular and singular torus characters in GL_3(F_q)

By [F1] every principal series of ${\mathrm{GL}}_3({\mathbb{F}}_q)$ has dimension $[G:B]=(q+1)(q^2+q+1)$.

Case (a): if the three coordinates are pairwise distinct, no nontrivial permutation fixes the character, so $W_{\chi }=1$; by [F2] $\dim {\mathrm{End}}_G(I(\chi ))=1$ and $I(\chi )$ is irreducible.

Case (b): if $\chi =(a,a,b)$ with $a\mathrm{\ne }b$, the stabiliser is $W_{\chi }\cong S_2$, so $\dim {\mathrm{End}}_G(I(\chi ))=2$ by [F2]. The equal-coordinate blocks are $n_1=2,n_2=1$, so by [F3] the constituents are indexed by the tuples $({\lambda }^{(1)},{\lambda }^{(2)})$ with ${\lambda }^{(1)}⊢2$, ${\lambda }^{(2)}⊢1$, and have multiplicities $f^{{\lambda }^{(1)}}{f}^{{\lambda }^{(2)}}$; by [F4] these multiplicities are all $1$, so there are exactly two non-isomorphic constituents, each of multiplicity one, and ${\mathrm{End}}_G(I(\chi ))\cong \mathbb{C}\oplus \mathbb{C}$.

Put $\psi =a\circ \det$ and let $W=\mathbb{C}{z}_0$ with $g{z}_0=\psi (g)z_0$. Multiplicativity and the triangular determinant formula show that $\psi$ is a character and $\psi {\mathrm{\mid }}_B$ is the inflation of $(a,a,a)$ (For same-sized finite square matrices over a commutative ring, $\det (AB)=\det (A)\det (B)$, The determinant of a triangular matrix is the product of its diagonal entries). Define $\mathrm{\Phi }:I(1)\otimes W\to I(a,a,a)$ by $\mathrm{\Phi }(f\otimes z_0)(x)=f(x)\psi (x{)}^{-1}$. This lands in $I(a,a,a)$ because $f(xb)=f(x)$ and $\psi (xb)=\psi (x)\psi (b)$, and it is $G$-equivariant since $\mathrm{\Phi }(g\cdot (f\otimes z_0))(x)=f(g^{-1}x)\psi (g)\psi (x{)}^{-1}=\mathrm{\Phi }(f\otimes z_0)(g^{-1}x)$ by [F6]. Its inverse sends $F$ to $(x\mapsto F(x)\psi (x))\otimes z_0$, whose function is right $B$-invariant. Thus $I(a,a,a)\cong I(1)\otimes (a\circ \det )$.

Case (c): if $\chi =(a,a,a)$, then $W_{\chi }=S_3$ and $\dim {\mathrm{End}}_G(I(\chi ))=6$ by [F2]; by [F3] the constituents are indexed by the partitions $\lambda ⊢3$ with multiplicities $f^{\lambda }$, which by [F4] are $1,2,1$ for $(3),(2,1),(1,1,1)$, and ${\mathrm{End}}_G(I(\chi ))\cong \mathbb{C}\times {\mathrm{M}}_2(\mathbb{C})\times \mathbb{C}\cong \mathbb{C}[S_3]$. Moreover the twist isomorphism of step 1.4 with $\psi =a\circ \det$ gives $I(a,a,a)\cong I(1)\otimes (a\circ \det )$; tensoring with a one-dimensional character preserves dimensions and multiplicities, so the constituent count and multiplicities agree with the spherical computation [F5] after twisting.

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 give the dimension, the three cases (a)-(c) with their endomorphism dimensions and constituent multiplicities, and the agreement with the general corollary and with the twisted spherical computation in case (c). AC is carried only from the Tits-deformation supplier in [F7], as declared; all modules are finite-dimensional over $\mathbb{C}$.