Let k be a finite field, G the k-points of GL\_2, T1, T2 the k-points of the split and non-split tori of G. 


Then the G-representations C[G/T1] and C[G/T2] are almost the same. 
More precisely, they differ by two copies of a certain irreducible representation (the Steinberg). I might have slightly miscomputed, but the point is that the decomposition of C[G/T1] and C[G/T2] into irreducibles is much more similar than what you might naively expect. 


Question: Is there a general phenomenon, of which this is a special case? 


Edit added: it seems like a corresponding alternating sum over tori for GL\_3 might be six copies of the Steinberg, see comment below. 



Induction from split and non-split tori  for GL_2 over a finite field
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. 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(Humphreys proved it for $G$ simply connected, semisimple, split algebraic groups over finite fields and Jantzen (unpublished)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 $(St\_G+1)+(St\_G-1) = 2St\_G$.


