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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.

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

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. 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?