Complex semisimple algebraic group, Borel, and flag variety

Assume the Axiom of Choice (The Axiom of Choice). Throughout this page, $G$ is a connected simply connected complex semisimple affine algebraic group: an affine group scheme $G$ of finite type over $\mathbb{C}$ whose underlying scheme is connected and smooth, whose Lie algebra $\mathfrak{g}=\mathrm{Lie}G$ is a semisimple complex Lie algebra, and which is simply connected in the sense that every central isogeny $G^{\mathrm{\prime }}\to G$ of connected affine algebraic groups over $\mathbb{C}$ with finite kernel is an isomorphism. These conditions are hypotheses on $G$, fixed once and for all; the local construction of the root subgroups, of $B$ and of $G\mathrm{/}B$ below is where they are used.

Maximal torus, roots, positive roots. Fix a maximal torus $T\subseteq G$, that is, a closed subgroup isomorphic to a product of copies of ${\mathbb{G}}_m$ which is maximal for this property. Its Lie algebra $\mathfrak{h}=\mathrm{Lie}T$ is a Cartan subalgebra of $\mathfrak{g}$ and $\mathfrak{g}$ decomposes as $\mathfrak{g}=\mathfrak{h}\oplus {⨁}_{\alpha \in \mathrm{\Phi }}{\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{\alpha }$ the root space of the root $\alpha \in {\mathfrak{h}}^{\ast }$ in the sense of Root and root space. We write $\mathrm{\Phi }=\mathrm{\Phi }(G,T)\subseteq {\mathfrak{h}}^{\ast }$ for this root set, a reduced crystallographic root system in the real span of $\mathrm{\Phi }$, and we fix once and for all a positive system ${\mathrm{\Phi }}^{+}$ with simple roots $\mathrm{\Delta }$, in the sense of Positive systems and simple roots. Define the nilpotent Lie subalgebras ${\mathfrak{n}}^{\pm }={⨁}_{\alpha \in {\mathrm{\Phi }}^{\pm }}{\mathfrak{g}}_{\alpha },\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}\mathrm{.}$ Here $\mathfrak{b}$ is the Borel subalgebra of Positive and negative nilpotent subalgebras and the Borel.

Borel subgroup and unipotent radical. Reserve $B\subseteq G$ for the closed connected subgroup with $\mathrm{Lie}B=\mathfrak{b}$ constructed in Borel, opposite unipotent groups and root coordinates, and $U\subseteq B$ for its unipotent radical. That lemma proves $B=T⋉U$ and identifies $U$ with the product of the positive-root subgroups. Reserve $U_{\alpha }$ for the closed one-parameter subgroup with $\mathrm{Lie}{U}_{\alpha }={\mathfrak{g}}_{\alpha }$ constructed, with its $T$-equivariance, in Algebraic root subgroups from root exponentials. These symbols name the later constructions; this definition does not establish their existence.

Weyl group. Let $N_G(T)$ be the normalizer subgroup scheme of $T$ in $G$, and put $W=N_G(T)\mathrm{/}T\mathrm{.}$ Here the quotient means the fppf quotient sheaf. Its identification with the constant algebraic group of the abstract Weyl group $W(\mathrm{\Phi })=⟨s_{\alpha }:\alpha \in \mathrm{\Phi }⟩$ of Weyl group remains a proof obligation for the later root-representative and Bruhat constructions. Until then $W$ has its action on $\mathfrak{h}$ and on $X^{\ast }(T)$ by conjugation. The formula does not identify $W(R)$ with $N_G(T)(R)\mathrm{/}T(R)$ for an arbitrary test algebra $R$.

Flag variety. Reserve $X=G\mathrm{/}B$ for the projective homogeneous $G$-variety constructed in Projective orbit constructions for G/B and G/P_alpha as the orbit of the highest-weight line $v_B={\wedge }^{\dim \mathfrak{b}}\mathfrak{b}$ in the Plücker representation ${\wedge }^{\dim \mathfrak{b}}\mathfrak{g}$. Once constructed, $X(R)$ is the set of $R$-points of that closed orbit for a $\mathbb{C}$-algebra $R$. The represented quotient functor is the fppf sheafification of the presheaf $R\mapsto G(R)\mathrm{/}B(R)$; a given $R$-point lifts to $G(R)$ precisely when its pulled-back $B$-torsor is trivial. The quotient identification and Zariski local sections of $G\to X$ are established in Zariski sections of Borel and minimal-parabolic orbit maps and A semisimple flag variety is smooth and projective.

Minimal parabolic. For a simple root $\alpha \in \mathrm{\Delta }$, reserve $P_{\alpha }$ for the subgroup generated by $B$ and the negative root subgroup $U_{-\alpha }$. The later lemma Minimal parabolic from one negative simple root proves that it is closed, contains $B$ and $U_{-\alpha }$, has $\mathrm{Lie}{P}_{\alpha }=\mathfrak{b}\oplus {\mathfrak{g}}_{-\alpha }$, and satisfies $P_{\alpha }=B\bigsqcup B{n}_{\alpha }B$ with $n_{\alpha }$ a Weyl representative of $s_{\alpha }$. That lemma and A minimal-parabolic flag projection is a projective-line bundle prove $P_{\alpha }\mathrm{/}B\cong {\mathbb{P}}^1$ and that $G\mathrm{/}B\to G\mathrm{/}{P}_{\alpha }$ is a Zariski locally trivial ${\mathbb{P}}^1$-bundle.

Conventions. All schemes and algebraic groups in this definition and in every item that depends on it are over $\mathbb{C}$. The Axiom of Choice is assumed and is used only through the published Lie-theoretic suppliers of the root data and through the injective-resolution and sheaf-cohomology supplies named by the individual items; the finite-type affine-group lemma A finite-type affine algebraic group has a faithful rational representation is choice-free.

This is the minimal parabolic strictly containing $B$, rather than the maximal-parabolic convention of Compositions, partial flags, and standard parabolics ↗. The later construction gives $\dim (G\mathrm{/}{P}_{\alpha })=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1$.