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

Embed the space $\mathcal{B}$ of complete isotropic flags intoThe incidence conditions $F_i\subset F_{i+1}$ and isotropy equations $⟨F_i,F_i⟩=0$ are closed polynomial conditions. Since the Grassmannian is projective, $\mathcal{B}$ is a projective algebraic variety.

Every complete isotropic flag extends to a symplectic basis. A symplectic change of basis carries any such flag to any other, so $G={\mathrm{Sp}}_{2n}$ acts transitively. The stabilizer $B$ of the standard flag consists of the upper triangular symplectic matrices. It is closed, connected, and solvable. The Lie-Kolchin theorem shows that every connected solvable subgroup fixes a complete flag in $V$; preservation of the symplectic form makes the resulting flag isotropic after taking its first half. Such a subgroup is conjugate into $B$, so $B$ is maximal and hence a Borel subgroup.

Solved by gpt-5.6-sol high.