The Weyl group, Borel subgroups and chambers

The normalizer quotient $W=N_G(T)\mathrm{/}T$ is finite étale by the structural root Definition. Its action on $X(T)$ is faithful: after algebraic closure, a normalizer element acting trivially on the character lattice centralizes $T$, and then lies in $C_G(T)=T$ by [F4]. The exact root-subgroup theorem F1 identifies its geometric group with the reflection group, and F1 supplies a $k$-rational normalizer representative for every root reflection. Products of these representatives therefore realize every geometric element of $W$ over $k$. A finite étale scheme all of whose geometric points are rational is a disjoint union of copies of $\mathrm{Spec}k$, so $W$ is constant. This proves constantness from the actual representatives, rather than from torsor lifting alone.

Let $T^{\mathrm{\prime }}$ be any split maximal torus. Then $(G,T^{\mathrm{\prime }})$ is a split reductive pair, and there is an integral regular cocharacter ${\lambda }^{\mathrm{\prime }}$: outside finitely many rational hyperplanes in $X_{\ast }(T^{\mathrm{\prime }})\otimes \mathbb{R}$ choose a rational point and clear denominators. If there are no roots, the weight decomposition gives $\mathrm{Lie}G=\mathrm{Lie}{T}^{\mathrm{\prime }}$, hence $G=T^{\mathrm{\prime }}$ by smoothness and connectedness, and it is its own Borel. Otherwise [F2], applied to $(G,T^{\mathrm{\prime }})$, makes $P_G({\lambda }^{\mathrm{\prime }})$ a Borel over $k$ containing $T^{\mathrm{\prime }}$. For every Borel $B\supseteq T$, [F2] gives $B=P_G(\lambda )$ for a regular $k$-cocharacter $\lambda$, and [F5] gives $B=U⋊T$ with $U=R_u(B)$.

For a field extension $k^{\mathrm{\prime }}\mathrm{/}k$, constantness in step 1.1 identifies $W(k^{\mathrm{\prime }})$ with the same finite group as $W(k)$. Every element therefore has a $k$-rational normalizer representative, constructed in step 1.1 as a product of the root-reflection representatives supplied by F1. Base change of this representative gives a point of $N_G(T)(k^{\mathrm{\prime }})$ mapping to the specified element of $W(k^{\mathrm{\prime }})$. Two such points have the same image precisely when their quotient lies in $T(k^{\mathrm{\prime }})$, since the scheme-theoretic kernel is $T$. Thus the quotient map induces the bijection $N_G(T)(k^{\mathrm{\prime }})\mathrm{/}T(k^{\mathrm{\prime }})=W(k^{\mathrm{\prime }})$, completing (a).

By F1, the faithful action of $W$ on $X(T)$ is precisely the group generated by the root reflections, canonically $W(R)$. The root combinatorics [F3] show that the simple reflections for any base generate it. This proves(d) directly from the supplier, without a backward reference to a later step or an unrelated claim about generating $G$.

To prove splitness, put $m_{\alpha }=⟨\alpha ,\lambda ⟩>0$ for $\alpha \in {\mathrm{\Phi }}^{+}(B)$. In the ordered root coordinates of [F1], let $U_r$ be the product of the root groups with $m_{\alpha }\ge r$, for integers $r\ge 1$. This root subset is quasi-closed, since every root $i\alpha +j\beta$ with $i,j>0$ has weight $i{m}_{\alpha }+j{m}_{\beta }\ge r$. Thus [F1] makes each $U_r$ a smooth connected $T$-stable subgroup, with $U_1=U$ and $U_r=1$ for large $r$. For roots $\alpha ,\beta \in {\mathrm{\Phi }}^{+}$, every root coordinate of the commutator $[u_{\alpha }(x),u_{\beta }(y)]$ is a polynomial in $x,y$, is homogeneous for the $\lambda$-weights, and vanishes when either variable is zero. Each nonzero monomial $x^a{y}^b$ therefore has $a,b\ge 1$ and weight $a{m}_{\alpha }+b{m}_{\beta }\ge m_{\alpha }+m_{\beta }$. Conjugating a root generator of $U_r$ by a root generator of $U$ therefore keeps it in $U_r$, as does inverse conjugation, proving normality under $U$; $T$ preserves the weight subsets. Applying the same argument to $U_{r+1}$, the root-generator commutators vanish modulo this normal subgroup, so $[U,U_r]\subseteq U_{r+1}$. Thus $U_r$ is normal in $B=U⋊T$. Root coordinates of weight exactly $r$ induce $U_r\mathrm{/}{U}_{r+1}\cong {\prod }_{m_{\alpha }=r}{\mathbf{G}}_a$: ordering those factors first gives a variety product with $U_{r+1}$, and collecting root factors modulo $U_{r+1}$ adds those coordinates, since their commutators have greater weight. The coordinate projection is a surjective group morphism with kernel $U_{r+1}$. Refining these vector-group quotients by coordinate subspaces yields a normal series with ${\mathbf{G}}_a$ quotients; the split torus quotient $B\mathrm{/}U=T\cong {\mathbf{G}}_m^{\dim T}$ yields ${\mathbf{G}}_m$ quotients. Hence $B$ is split as a solvable group. Together with step 1.2 this proves (b).

The opposition theorem [F2] identifies Borels containing $T$ with the sign chambers of regular cocharacters, equivalently with positive root systems. Conjugation by a normalizer representative permutes root groups by [F1], hence transports these sign systems by its lattice action. Under the canonical identification in step2.2, [F3] gives the simple transitivity of that action and the bijection onto positive systems. This proves(c).

For (e), changing a representative $n_w$ by an element of $T(k)\subseteq B(k)$ preserves $B{n}_{w}B$. Conjugation by $n_w$ carries ${\mathfrak{g}}_{\alpha }$ to ${\mathfrak{g}}_{w\alpha }$, and its conjugate root group is smooth, connected and $T$-stable. The root-group containment criterion of F1, applied to that conjugate group and $U_{w\alpha }$, gives containment; both groups have dimension one, so they are equal. Thus $n_w{U}_{\alpha }{n}_w^{-1}=U_{w\alpha }$ scheme-theoretically. This proves (e).