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

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.

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Choose the maximal torusand write ${\epsilon }_i(t)=t_i$. Thenwhere the type-$C_n$ root system isHere $X^{\ast }(T)={⨁}_i\mathbb{Z}{\epsilon }_i$ and $X_{\ast }(T)={⨁}_i\mathbb{Z}{\epsilon }_i^{\vee }$. The root datum haswith the corresponding negatives.

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The normalizer of $T$ permutes the $n$ symplectic coordinate planes and may interchange the two basis vectors in each plane. Modulo $T$, these operations give every signed permutation of $({\epsilon }_1,\dots ,{\epsilon }_n)$. Thereforethe signed symmetric group, generated by adjacent coordinate transpositions and one sign change.

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For each signed permutation $w\in W$, choose its symplectic signed-permutation matrix $\dot{w}\in N_G(T)$. The Bruhat decomposition of a reductive algebraic group is the explicit disjoint union

To prove existence, compare the standard isotropic flag $F_{\bullet }$ with $g{F}_{\bullet }$. The rankstogether with the symplectic orthogonality relations determine a unique signed permutation $w$. Symplectic row and column operations from $B$ then reduce $g$ to $\dot{w}$, so $g\in B\dot{w}B$. Conversely, the same intersection dimensions are constant on a double coset and recover $w$, proving disjointness. This is symplectic Gaussian elimination and establishes the claimed decomposition.

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Put $i_0=0$ and $d_j=i_j-i_{j-1}$. The stabilizeris the standard parabolic subgroup containing $B$ obtained by allowing arbitrary changes of basis inside the successive blocks. Its Levi subgroup isThe 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 $F_{i_r}^{\perp }/F_{i_r}$. Hencethe 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.

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The center consists of scalar symplectic transformations:This description retains the nonreduced center in characteristic two. Since $Z_G$ is a finite central subgroup scheme, the invariant ring $k[G{]}^{Z_G}$ is finitely generated andis an affine algebraic group; the quotient map is finite and faithfully flat.

The quotient torus is $T/{\mu }_2$. Its character and cocharacter lattices areThe roots and coroots are the same type-$C_n$ sets written in part (ii), now regarded in these lattices. This is the adjoint root datum of type $C_n$.

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