Projective orbit constructions for G/B and G/P_alpha

The orbit maps. By [F7] the actions of $G$ on $W_B$ and $W_{\alpha }$ are rational representations, hence morphisms $G\to GL(W_B)$, $G\to GL(W_{\alpha })$; the induced actions on the projective spaces are morphisms and ${\pi }_B$, ${\pi }_{\alpha }$ are morphisms of varieties with $G$-stable images. By [F1] the scheme $G$ is connected and smooth of finite type over the algebraically closed field $\mathbb{C}$, hence regular by [F12], so its local rings are domains and a connected such scheme is irreducible; thus $G$ is irreducible of dimension $\dim \mathfrak{g}$, and ${\pi }_B(G)$, ${\pi }_{\alpha }(G)$ are nonempty irreducible subsets whose closures $X_B$, $X_{\alpha }$ are nonempty irreducible closed subvarieties. Both are $G$-stable: for $g,h\in G$ one has $g\cdot h[v_B]=gh[v_B]$.

Weight of a transported highest weight vector. Let $w\in W$ with representative $n_w\in N_G(T)$ and let $t\in T$. Using the $B$-stability of the line $\mathbb{C}{v}_B$ with $T$-weight $2\rho$ of [F7], and writing $(w\lambda )(t)=\lambda (n_w^{-1}t{n}_w)$ for the action of $W$ on characters of $T$ of [F5], one computes in the module $W_B$ $t\cdot (n_w{v}_B)=(t{n}_w)\cdot v_B=n_w\cdot ((n_w^{-1}t{n}_w)\cdot v_B)=2\rho (n_w^{-1}t{n}_w)n_w{v}_B=(w2\rho )(t)n_w{v}_B\mathrm{.}$ So $n_w{v}_B$ is a nonzero weight vector of weight $w(2\rho )$; the same computation in $W_{\alpha }$ gives weight $w(2\rho -\alpha )$ for $n_w{v}_{\alpha }$.

Positive root factors preserve the initial weight component. By [F2] write $u\in U$ as a finite product of ${\exp }_G(z_{\beta }{e}_{\beta })$. For a weight vector $v$ of weight $\mu$, the representation identity gives $H(e_{\beta }v)=\mu (H)e_{\beta }v+[H,e_{\beta }]v=(\mu +\beta )(H)e_{\beta }v$. Thus $d\varrho (e_{\beta })$ raises weights by $\beta$, and is nilpotent because the module has finitely many weights. By [F10] the corresponding root factor acts as the finite polynomial ${\sum }_{k\ge 0}{z}_{\beta }^{k}d\varrho (e_{\beta }{)}^k\mathrm{/}k!$. Expanding the finite product, its constant term sends $v$ to $v$; every other nonzero term has weight $\mu +{\sum }_{\beta }{k}_{\beta }\beta$ with nonnegative integers $k_{\beta }$, at least one positive. A nonempty sum of positive roots is nonzero (evaluate on a vector defining the positive system). Hence the weight-$\mu$ component of $uv$ is exactly $v$. This uses each exponential separately and requires no global logarithm.

The stabilizer of $[v_B]$ is $B$. Since $\mathbb{C}{v}_B$ is $B$-stable by [F7], $B\subseteq H_B$. Conversely let $g\in G$ with $g[v_B]=[v_B]$. By the union and cell parametrization of [F4] there are $w\in W$ and a factorization $g=u{n}_{w}b$ with $u\in U_w\subseteq U$ and $b\in B$. Let $\chi :B\to {\mathbb{G}}_m$ be the character with $b^{\mathrm{\prime }}{v}_B=\chi (b^{\mathrm{\prime }})v_B$ for $b^{\mathrm{\prime }}\in B$, which exists because the line is $B$-stable. Then $g{v}_B=\chi (b)u(n_w{v}_B),$ whose component in the weight $w(2\rho )$ is $\chi (b)n_w{v}_B\mathrm{\ne }0$ by steps 1.2 and 1.3, while $g{v}_B\in {\mathbb{C}}^{\times }{v}_B$ lies in the weight space of weight $2\rho$ of [F7]. If $w\mathrm{\ne }1$ then $w(2\rho )\mathrm{\ne }2\rho$ by [F9], because $2\rho$ is strictly dominant and hence has trivial stabilizer in $W$; distinct weight spaces are independent by [F8], so a nonzero vector with a nonzero component in weight $w(2\rho )\mathrm{\ne }2\rho$ cannot lie in $\mathbb{C}{v}_B$. Therefore $g{v}_B\in {\mathbb{C}}^{\times }{v}_B$ forces $w=1$, and then $g=ub\in B$ because $n_1=1$ and $u\in U\subseteq B$. Hence $H_B=B$.

The orbits are closed. Let $O_B={\pi }_B(G)$. By [F14] this is a constructible subset of its irreducible closure $X_B$ and contains a nonempty open subset $V_B\subseteq X_B$. For every $x\in O_B$, choose $g\in G$ carrying a point of $V_B$ to $x$; such a $g$ exists by transitivity of the $G$-action on the orbit. The translate $g{V}_B$ is open in $X_B$, lies in $O_B$, and contains $x$. Thus $O_B$ itself is open in $X_B$, and its complement ${\mathrm{\partial }}_B=X_B\setminus O_B$ is closed and $G$-stable. If ${\mathrm{\partial }}_B$ were nonempty, [F13] applied to this reduced closed projective $B$-stable subvariety would give a $B$-fixed point $y\in {\mathrm{\partial }}_B$. Its line is $B$-stable, so [F7] forces $y=[v_B]\in O_B$, a contradiction. Hence $X_B=O_B$. The same constructible-open and transitivity argument makes $O_{\alpha }={\pi }_{\alpha }(G)$ open in $X_{\alpha }$; its complement is closed and $B$-stable, and [F13] with the unique $B$-stable line of [F7] makes it empty. Consequently both orbits are closed and the orbit maps surject onto $X_B,X_{\alpha }$.

The stabilizer of $[v_{\alpha }]$ is $P_{\alpha }$. The subgroup $P_{\alpha }=B\bigsqcup B{n}_{\alpha }B$ of [F6] stabilizes the line $\mathbb{C}{v}_{\alpha }$: this holds for $B$ by [F7], and $n_{\alpha }{v}_{\alpha }$ has weight $s_{\alpha }(2\rho -\alpha )=2\rho -2\alpha +\alpha =2\rho -\alpha$, the highest weight of the irreducible module $L(2\rho -\alpha )$ of [F7], whose weight space is one-dimensional by [F8] and [F7]; since $n_{\alpha }$ acts invertibly and $v_{\alpha }\mathrm{\ne }0$, $n_{\alpha }{v}_{\alpha }$ is a nonzero element of that one-dimensional space, so $n_{\alpha }{v}_{\alpha }\in {\mathbb{C}}^{\times }{v}_{\alpha }$. Hence $P_{\alpha }\subseteq H_{\alpha }$. Conversely let $g[v_{\alpha }]=[v_{\alpha }]$ and write $g=u{n}_{w}b$ as in step 2.1. The same component computation as in step 2.1, with $2\rho -\alpha$ in place of $2\rho$ and using $n_{\alpha }$ in place of $n_w$ for the four terms $(s_{\alpha }(2\rho -\alpha ))=2\rho -\alpha$ of [F5], shows that the component of $g{v}_{\alpha }$ in the weight $w(2\rho -\alpha )$ equals ${\chi }_{\alpha }(b)$ times the nonzero vector $n_w{v}_{\alpha }$, where ${\chi }_{\alpha }$ is the character by which $B$ acts on $\mathbb{C}{v}_{\alpha }$; since $g{v}_{\alpha }\in {\mathbb{C}}^{\times }{v}_{\alpha }$ has pure weight $2\rho -\alpha$, this forces $w(2\rho -\alpha )=2\rho -\alpha$. By [F9] the stabilizer of the dominant weight $2\rho -\alpha$ in $W(\mathrm{\Phi })$ is generated by the simple reflections $s_i$ with $⟨2\rho -\alpha ,{\alpha }_i^{\vee }⟩=0$, which by [F9] is exactly $\{1,s_{\alpha }\}$. So $w\in \{1,s_{\alpha }\}$ and $g\in B\cup B{n}_{s_{\alpha }}B=B\cup B{n}_{\alpha }B=P_{\alpha }$ by [F4], [F5] and [F6]. Hence $H_{\alpha }=P_{\alpha }$.

Fibres of the orbit maps. For $g_1,g_2\in G$ one has ${\pi }_B(g_1)={\pi }_B(g_2)$ if and only if $g_1^{-1}{g}_2\in H_B=B$, that is $g_2\in g_{1}B$; thus the fibres of ${\pi }_B$ over points of ${\pi }_B(G)$ are exactly the right cosets of $B$, each isomorphic to $B$ by translation and of dimension $\dim B$. The same argument with $H_{\alpha }=P_{\alpha }$ shows that the fibres of ${\pi }_{\alpha }$ over points of ${\pi }_{\alpha }(G)$ are exactly the right cosets of $P_{\alpha }$, of dimension $\dim P_{\alpha }=\dim B+1$ by [F6].

Dimensions. Apply [F11] to the morphism ${\pi }_B:G\to \mathbb{P}(W_B)$ with $G$ irreducible by step 1.1: the closure $X_B$ is irreducible, $\dim G=\dim X_B+r$, and $r$ is the common dimension of the nonempty fibres over a nonempty open subset of $X_B$; by step 4.1 all fibres over points of ${\pi }_B(G)$ have dimension $\dim B$, so $r=\dim B$ and $\dim X_B=\dim G-\dim B=\dim \mathfrak{g}-\dim \mathfrak{b}=(\dim \mathfrak{h}+2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid })-(\dim \mathfrak{h}+\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid })=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ by [F1] and [F2]. Applying the same argument to ${\pi }_{\alpha }$ and using $\dim P_{\alpha }=\dim B+1$ gives $\dim X_{\alpha }=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1$.

Smoothness and connectedness. By steps 1.1, 5.1 and 2.2 the reduced closed subvariety $X_B\subseteq \mathbb{P}(W_B)$ is nonempty, irreducible and of dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, and $G$ acts on $X_B$ by automorphisms, transitively on its point set because $X_B={\pi }_B(G)$. Hence [F12] shows that $X_B$ is regular, and, $\mathbb{C}$ being perfect, that $X_B$ is smooth over $\mathbb{C}$; the same holds for $X_{\alpha }$. Irreducibility gives connectedness, and $X_B$, $X_{\alpha }$ are closed subvarieties of projective spaces, hence projective.

Conclusion. Steps 2.1 and 3.1 give the stabilizer statement (ii); steps 4.1, 5.1 and 2.2 give the fibre statement and the dimensions of (iii) and (i); step 6.1 gives smoothness, connectedness and projectivity, and the orbit maps induce the bijections $G\mathrm{/}B\to X_B$, $G\mathrm{/}{P}_{\alpha }\to X_{\alpha }$ on orbit spaces, exhibiting $X_B$ and $X_{\alpha }$ as the quotients $G\mathrm{/}B$ and $G\mathrm{/}{P}_{\alpha }$. The Axiom of Choice is assumed in the statement and declared as the dependency The Axiom of Choice; it is inherited through the suppliers [F10], [F11], [F12] and [F13] and through the highest weight classification used in [F7]; the argument itself selects only the fixed simple root $\alpha$, the finitely many root representatives $n_w$ used in steps 2.1 and 3.1, and the character $\chi$ of [F7].