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

Table of contents
- Solution i

Solution

The Mackey restriction formula says that for subgroups

H, K \leq G

and a

K

-representation

W

,

Res_{H}^{G} Ind_{K}^{G} W ≅ ⨁_{x \in H \ G / K} Ind_{H \cap x K x^{- 1}}^{H} Res_{H \cap x K x^{- 1}}^{x K x^{- 1}} (^{x} W),

where

^{x} W (x k x^{- 1}) = W (k)

.

For

H = K = B

, the Bruhat decomposition of SL2 over a finite field gives

G = B ⊔ BwB, w = (01 - 1 0) .

Thus

B \ G / B

has representatives

1, w

. The identity double coset contributes

θ

, while

B \cap wB w^{- 1} = T

contributes an induced representation from the diagonal subgroup. Therefore

Res_{B}^{G} Ind_{B}^{G} θ ≅ θ \oplus Ind_{T}^{B} (θ^{w} ∣_{T}),

where

θ^{w} (t) = θ (w^{- 1} tw) .

For

t (a) = diag (a, a^{- 1})

, one has

w^{- 1} t (a) w = t (a^{- 1})

.

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