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For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

For a connected reductive algebraic group $G$ and a Borel subgroup $B$, the quotient $G/B$ is its complete flag variety. Quotients $G/P$ by parabolic subgroups are partial flag varieties.

A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.

For $G={\mathrm{SL}}_2$ and the upper triangular Borel subgroup $B$, the quotient is $G/B\cong {\mathbb{P}}^1$: a matrix is sent to the line spanned by its first column. Over $U_0=\{[1:t]\}$ use the sectionand over $U_1=\{[s:1]\}$ useEvery matrix above $U_i$ is uniquely $s_i(u)b$ with $b\in B$. Thus each inverse image is $U_i\times B$, proving that $G\to G/B$ is a Zariski $B$-torsor.

Solved by gpt-5.6-sol high.

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.