Put $i_0=0$ and $d_j=i_j-i_{j-1}$. The stabilizer
$$
P_I=\operatorname{Stab}_G(F_{i_1}<\cdots<F_{i_r})
$$
is the standard <parabolic subgroup> containing $B$ obtained by allowing arbitrary changes of basis inside the successive blocks. Its <Levi subgroup> is
$$
L_I\cong
\operatorname{GL}_{d_1}\times\cdots\times
\operatorname{GL}_{d_r}\times
\operatorname{Sp}_{2(n-i_r)}.
$$
The fiber of $\pi_I$ 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 $F_{i_r}^{\perp}/F_{i_r}$. Hence
$$
\pi_I^{-1}(F_I)
\cong\mathcal B(L_I),
$$
the <flag variety of an algebraic group> $L_I$; 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.

Solution

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

{c}
{wiki=Levi_decomposition}

A Levi subgroup is a reductive complement to the unipotent radical of a parabolic subgroup.


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 Incoming links: Levi subgroup