Induction from split and non-split tori for GL_2 over a finite field

Question

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.

Answer 1

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(Humphreys proved it for $G$ simply connected, semisimple, split algebraic groups over finite fields and Jantzen (unpublished) generalised it to arbitrary reductive groups over finite fields.)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 $(St_G+1)+(St_G-1) = 2St_G$.

Answer 2

Certainly representations associated to the different tori of G play an important role in Deligne-Lusztig theory. Unfortunately, the whole point of D-L theory is that if you want to understand the representation theory of the finite field points of algebraic groups, you should still remember that they are algebraic varieties, and replace naive pushforwards in the category of sets with ones in the etale topology.

I suspect what you're seeing is that the representation theory of $GL_2$ is controlled by a very simple Coxeter group, the symmetric group on 2 letters $S_2$, and the generalization will be a lot uglier.