Bruhat double cosets from rank-one multiplication

Conjugation preserves root subgroups in the needed algebraic sense. If $n\in N_G(T)(\mathbb{C})$ acts on characters by $\sigma$, then $\mathrm{Ad}(n){\mathfrak{g}}_{\beta }={\mathfrak{g}}_{\sigma \beta }$ by the defining root-space eigenvalue equation. The target is one-dimensional; hence $\mathrm{Ad}(n)e_{\beta }=c{e}_{\sigma \beta }$ for some $c\mathrm{\ne }0$. Apply exponential naturality [F8] to the conjugation automorphism $C_n$ and the explicit curves [F2]: $n{u}_{\beta }(z)n^{-1}=u_{\sigma \beta }(cz)$ for every $z\in \mathbb{C}$. Thus $n{U}_{\beta }{n}^{-1}=U_{\sigma \beta }$ as closed subgroup schemes: both morphisms are algebraic and agree on the reduced affine line's $\mathbb{C}$-points. This use of exponentials is within the present characteristic-zero complex-group scope.

If $n\in N_G(T)(\mathbb{C})$ preserves ${\mathrm{\Phi }}^{+}$, then $n$ normalizes $U$, $U^{-}$ and $B$ by step 1.1 and [F3]. The open subsets $\mathrm{\Omega }=U^{-}B$ and $\mathrm{\Omega }n=U^{-}nB$ of the irreducible $G$ meet by [F1]. At an intersection write $u_1^{-}{b}_1=u_2^{-}n{b}_2$; rearranging puts $n=(u_2^{-}{)}^{-1}{u}_1^{-}{b}_1{b}_2^{-1}\in \mathrm{\Omega }$. Write its unique big-cell coordinates as $n=u^{-}tu$ and put $ns=\sigma (s)n$ for $s\in T$. The big-cell coordinates of $ns=u^{-}(ts)(s^{-1}us)$ and $\sigma (s)n=(\sigma (s)u^{-}\sigma (s{)}^{-1})(\sigma (s)t)u$ have middle factors $ts$ and $\sigma (s)t$. Uniqueness gives $\sigma (s)=s$ for every $s$, so $n$ centralizes $T$. Comparing the outer coordinates again gives $s^{-1}us=u$ and $s{u}^{-}{s}^{-1}=u^{-}$ for every $s\in T$. In the polynomial root coordinates [F3], conjugation by $s$ scales the coordinate indexed by $\beta$ by $\beta (s)$; since no root character is trivial, all coordinates vanish. Hence $u=u^{-}=1$ and $n=t\in T$. In particular $C_G(T)(\mathbb{C})=T(\mathbb{C})$ and $B(\mathbb{C})\cap N_G(T)(\mathbb{C})=T(\mathbb{C})$: the latter follows also directly by comparing the unique $T⋉U$ coordinates of $bs$ and $\sigma (s)b$.

Every $n\in N_G(T)(\mathbb{C})$ permutes the root set by step 1.1 and carries ${\mathrm{\Phi }}^{+}$ to a positive system. By simple transitivity [F6], there is a unique abstract $w\in W(\mathrm{\Phi })$ carrying ${\mathrm{\Phi }}^{+}$ to this system. Choose a simple-reflection word for $w$ and multiply its rank-one representatives [F4] to obtain $n_w$ with the same action on roots. Then $n_w^{-1}n$ preserves ${\mathrm{\Phi }}^{+}$ and lies in $T$ by step 2.1. Conversely the $n_{\alpha }$ realize the simple reflections, so the map $N_G(T)(\mathbb{C})\mathrm{/}T(\mathbb{C})\to W(\mathrm{\Phi })$ is surjective and injective, and different words for $w$ differ by $T$. This proves the identification promised in the statement without assuming Coxeter relations for the representatives.

The scheme-theoretic quotient has the same finite set of components. Because $G$ is affine of finite type and $T$ closed, the condition $gT{g}^{-1}\subseteq T$ is closed in $g$: choose finite generators for the ideal of $T$ in $\mathcal{O}(G)$, pull each through conjugation $G\times T\to G$, and set to zero its finitely many coefficients in $\mathcal{O}(T)$; impose the analogous equations for $g^{-1}Tg\subseteq T$. Their intersection represents the normalizer functor $N=N_G(T)$ as a closed finite-type subgroup scheme. Differentiating the normalizing condition at the identity gives the inclusion $\mathrm{Lie}N\subseteq \{X\in \mathfrak{g}:[X,\mathfrak{h}]\subseteq \mathfrak{h}\}$. The root decomposition makes the set on the right equal to $\mathfrak{h}$, while $T\subseteq N$ gives the reverse inclusion $\mathfrak{h}\subseteq \mathrm{Lie}N$, so $\dim T=\dim \mathrm{Lie}N$; since $T\subseteq N$ is smooth of this dimension, the local ring of $N$ at the identity is regular, and group translations make $N$ smooth and reduced everywhere. By step 3.1 the closed cosets $n_{w}T$ exhaust $N(\mathbb{C})$ and are pairwise disjoint. A reduced finite-type $\mathbb{C}$-scheme is Jacobson, so its closed points are dense in every nonempty locally closed subset; the finite union of those closed cosets therefore equals $N$ as a scheme, and each coset is open as well as closed. On each component the quotient map $n_{w}T\to \mathrm{Spec}\mathbb{C}$ is the trivial right $T$-torsor. These components glue to a Zariski-locally trivial $T$-torsor $N\to {\coprod }_{w\in W(\mathrm{\Phi })}\mathrm{Spec}\mathbb{C}$, and the displayed target represents the fppf sheaf quotient $N\mathrm{/}T$. Multiplication agrees with $W(\mathrm{\Phi })$ on closed points, hence between the finite reduced constant schemes, proving (iv).

With the Weyl representatives established in step 3.1, fix a simple root $\alpha$, write $s=s_{\alpha }$, and let $U^{\mathrm{\prime }\mathrm{\prime }}$ be the subgroup generated by $U_{\beta }$ for $\beta \in {\mathrm{\Phi }}^{+}\setminus \{\alpha \}$. The root-coordinate and height-raising commutator law of [F3] gives $U=U^{\mathrm{\prime }\mathrm{\prime }}{U}_{\alpha }$: in a height-compatible order the simple $\alpha$ factor can be moved to the far right, because swapping it past another positive-root factor changes only factors at strictly larger heights, never a new $\alpha$ factor. Step 1.1 and the root-system fact that $s$ permutes ${\mathrm{\Phi }}^{+}\setminus \{\alpha \}$ show $n_{\alpha }{U}^{\mathrm{\prime }\mathrm{\prime }}{n}_{\alpha }^{-1}=U^{\mathrm{\prime }\mathrm{\prime }}$. Consequently, after absorbing torus and $U^{\mathrm{\prime }\mathrm{\prime }}$ factors into the left $B$, one has $n_{\alpha }B{n}_w\subseteq B{U}_{-\alpha }{n}_{\alpha }{n}_w$ for every $w$.

The product in step 4.2 occupies at most two cells, with a direct rank-one calculation. Take $n_{sw}=n_{\alpha }{n}_w$, permissible by step 3.1. If $w^{-1}\alpha >0$, then $n_{sw}^{-1}{U}_{-\alpha }{n}_{sw}=U_{w^{-1}\alpha }\subseteq B$ by step 1.1, so $B{U}_{-\alpha }{n}_{sw}\subseteq B{n}_{sw}B$. If $w^{-1}\alpha <0$, the zero parameter is in $B{n}_{sw}B$. For $z\mathrm{\ne }0$, direct multiplication in $S{L}_2$ gives $\left(\begin{array}{cc}1& 0\\ z& 1\end{array}\right)w=\left(\begin{array}{cc}-z^{-1}& -1\\ 0& -z\end{array}\right)\left(\begin{array}{cc}1& 0\\ -z^{-1}& 1\end{array}\right)$; applying ${\phi }_{\alpha }$, the first matrix lies in $B$, while $n_w^{-1}{U}_{-\alpha }{n}_w=U_{-w^{-1}\alpha }\subseteq B$, so $u_{-\alpha }(z)n_{\alpha }{n}_w\in B{n}_{w}B$. Therefore $n_{\alpha }B{n}_w\subseteq B{n}_{w}B\cup B{n}_{sw}B$ in both cases. This is Milne's two-cell inclusion, proved here from the displayed matrix identity and root coordinates.

Let $X={\bigcup }_{w\in W(\mathrm{\Phi })}B{n}_{w}B$. It contains $B$, is stable under left $B$ and right $B$, and is stable under left $U_{-\alpha }$ for each simple $\alpha$: the minimal-parabolic two-cell equality [F5] places $U_{-\alpha }$ inside $B\cup B{n}_{\alpha }B$, and step 5.1 controls $B{n}_{\alpha }B{n}_{w}B$. The subgroup $H$ generated by $B$ and the simple negative-root groups contains each $n_{\alpha }$, by the standard three-unipotent factorization of $w$ in $S{L}_2$ through [F4]. Every root is a Weyl translate of a simple root by [F6], so conjugation by products of the $n_{\alpha }$ and step 1.1 put every positive and negative root group in $H$. Thus $\mathrm{\Omega }=U^{-}B\subseteq H$ by [F1] and [F3]. Every left coset of $H$ contains an open translate of $\mathrm{\Omega }$, so every coset is open; connectedness of $G$ forces a single coset and $H=G$. Since $X$ contains $1$ and is stable under the generators of $H$, $X=G$. This proves coverage without asserting that an abstract generated subgroup is closed.

For each $w$ in the covering of step 6.1, put $I_w={\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}$, $J_w={\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{+}$. Both sets are closed under root addition: if $\beta ,\gamma$ are in either set and $\beta +\gamma$ is a root, its image under $w^{-1}$ has the same strict sign as the images of $\beta ,\gamma$. The corresponding sums ${\mathfrak{n}}_{I_w}={⨁}_{\beta \in I_w}{\mathfrak{g}}_{\beta }$ and ${\mathfrak{n}}_{J_w}={⨁}_{\beta \in J_w}{\mathfrak{g}}_{\beta }$ are Lie subalgebras, and the finite polynomial exponential/logarithm construction of [F3] makes their images $U_w$ and $U^w$ closed root-coordinate subgroups. In height-graded Lie coordinates, $\mathrm{BCH}(X,Y)=X+Y$ plus brackets of strictly greater height. Given $Z\in {\mathfrak{n}}^{+}$, solve $Z=\mathrm{BCH}(X,Y)$ recursively by height with $X\in {\mathfrak{n}}_{I_w}$ and $Y\in {\mathfrak{n}}_{J_w}$: in each root coordinate exactly one of $X,Y$ occurs linearly, while every bracket term uses already solved lower heights. The recursion is polynomial over $\mathbb{C}$ and gives a polynomial inverse to multiplication $U_w\times U^w\to U$ on every test algebra, hence a scheme isomorphism. By step 1.1, $n_w^{-1}{U}^w{n}_w\subseteq U$, and so $B{n}_{w}B=U{n}_{w}B=U_w{n}_{w}B$.

The parameter map $U_w\times B\to G$, $(u,b)\mapsto u{n}_{w}b$, is an isomorphism onto its image as a locally closed subscheme. Indeed, $V_w^{-}=n_w^{-1}{U}_w{n}_w$ is a closed root-coordinate subgroup of $U^{-}$ because $w^{-1}{I}_w\subseteq {\mathrm{\Phi }}^{-}$; left translation by $n_w^{-1}$ identifies the map with the restriction of the big-cell isomorphism $U^{-}\times B\stackrel{\sim }{\to }\mathrm{\Omega }$ of [F1] to the closed subscheme $V_w^{-}\times B$. Its image is therefore closed in the open $n_w\mathrm{\Omega }$, and step 7.1 identifies its underlying set with $B{n}_{w}B$. This proves (ii), including a regular inverse, rather than inferring an isomorphism from an injective differential.

The cells are disjoint. The chart in step 8.1 descends to $U_w\stackrel{\sim }{\to }B{n}_{w}B\mathrm{/}B$ because its second factor is the right $B$ action. The point $n_{w}B$ is fixed by $T$. In this chart the left $T$ action on $u{n}_{w}B$ sends $u$ to $tu{t}^{-1}$, since $n_w^{-1}t{n}_w\in T\subset B$. By [F3] every root coordinate of $U_w$ has a nontrivial $T$ weight, so $1$ is its unique $T$-fixed $\mathbb{C}$-point. If $n_{w^{\mathrm{\prime }}}B$ were in the $w$ cell, it would be fixed by $T$ and hence equal $n_{w}B$; then $n_w^{-1}{n}_{w^{\mathrm{\prime }}}\in B\cap N_G(T)=T$ by step 2.1, giving $w=w^{\mathrm{\prime }}$ by step 3.1. Any nonempty intersection of two double cosets contains a representative of each, so they are pairwise disjoint. Combined with step 6.1 this proves (i).

With the disjoint decomposition of step 9.1 established, [F7] gives $\mathrm{\mid }{I}_w\mathrm{\mid }=\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w)$, and the closed root-coordinate subgroup $U_w$ in step 7.1 is a product of $\mathrm{\mid }{I}_w\mathrm{\mid }$ copies of ${\mathbb{G}}_a$ as a variety. Hence $U_w\cong {\mathbb{A}}^{\mathrm{\ell }(w)}$ and $\dim U_w=\mathrm{\ell }(w)$, proving (iii). The Axiom of Choice is assumed and declared through The Axiom of Choice; its exact uses here are inherited from the exponential-naturalness supplier [F8], the root-factorization and big-cell suppliers [F1]–[F3], and the rank-one supplier [F4]. The finite Weyl representatives and the finite root orders require only finite choices.