I don't have much time, but maybe the following can lead you to a solution. I'm sloppy too, writing $G$ both for the algebraic group (over some finite field $k$) and for the set of points $G(k)$.
This is semilar to a special case of a formula of Humphreys on Deligne-Lusztig characters. ("Deligne-Lusztig characters and principal indecomposable modules". J. Algebra 62 (1980), no. 2, 299--303.) The special case says that
$\sum_{w \in W} R_{T_w}(1) = |W| St_G$,
where $W$ is the Weyl group and $T_w$ the rational maximal torus defined by $w$ (by twisting a fixed rational maximal torus $T$), and $St_G$ is the Steinberg character. In the simplest case, $T$ is split, and then the assignment $w \mapsto T_w$ induces a bijection between conjugacy classes in the Weyl group and rational conjugacy classes of rational maximal tori and conjugacy classes in the Weyl group. This is like the multiplicites you found for $GL_3$. Also signs come up, since the sign of the "dimension" of a DL representation is related to the parity of the split rank of the corresponding torus.
[Note that $R_T(1) = Ind_B^G(1)$ if $T$ is split and lies in the rational Borel $B$.]
(Humphreys proved it for $G$ simply connected, semisimple, split algebraic groups over finite fields and Jantzen (unpublished) generalised it to arbitrary reductive groups over finite fields.)
EDIT: $R_T(\theta)$ is the virtual representation of DL defined by the rational maximal torus $T$ and the character $\theta$ of the finite group $T(k)$.
For $GL_2$ the formula boils down to $(St_G+1)+(St_G-1) = 2St_G$.