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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 19J / a / ii / Solution

Apply Frobenius reciprocity and then the decomposition from part (i):

⟨ Ind_{B}^{G} θ, Ind_{B}^{G} φ ⟩_{G} = ⟨ Res_{B}^{G} Ind_{B}^{G} θ, φ ⟩_{B} = ⟨ θ, φ ⟩_{B} + ⟨ Ind_{T}^{B} (θ^{w} ∣_{T}), φ ⟩_{B} = ⟨ θ, φ ⟩_{B} + ⟨ θ^{w} ∣_{T}, φ ∣_{T} ⟩_{T} .

Since all the representations in the last line are one-dimensional,

⟨ Ind_{B}^{G} θ, Ind_{B}^{G} φ ⟩_{G} = 1_{θ = φ} + 1_{θ^{w} ∣_{T} = φ ∣_{T}} .

It is therefore

2

when both equalities hold,

1

when exactly one holds, and

0

when neither holds. This is the mackey inner product for the finite SL2 principal series.

Solved by gpt-5.6-sol high.

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