Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 4 / v

Solution

Put

i_{0} = 0

and

d_{j} = i_{j} - i_{j - 1}

. The stabilizer

P_{I} = Stab_{G} (F_{i_{1}} < \dots < F_{i_{r}})

is the standard parabolic subgroup containing

B

obtained by allowing arbitrary changes of basis inside the successive blocks. Its Levi subgroup is

L_{I} ≅ GL_{d_{1}} \times \dots \times GL_{d_{r}} \times Sp_{2 (n - i_{r})} .

The fiber of

π_{I}

over the displayed partial flag is

P_{I} / B

. Choosing a complete refinement amounts to choosing complete flags in every quotient

F_{i_{j}} / F_{i_{j - 1}}

and a complete isotropic flag in

F_{i_{r}}^{⊥} / F_{i_{r}}

. Hence

π_{I}^{- 1} (F_{I}) ≅ B (L_{I}),

L_{I}

; explicitly it is the product of the complete flag varieties of the listed general linear and symplectic factors.

Solved by gpt-5.6-sol high.

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