Beilinson–Bernstein Localization and Category O

Suppose $q=\kappa -w\kappa \in Q$. Since $q$ is real, $b=wb$, so $w\in W_b$. Move $b$ into the closed positive chamber and conjugate $a,b,\kappa ,w,q$ by the same Weyl element; lattice membership is preserved. With $I$ and $E_I$ as in [F2], $w\in W_I$, and each generator of $W_I$ fixes $E_I^{\perp }$. Therefore $a-wa\in E_I$; since $q=a-wa\in Q$, [F2] gives $q\in Q_I=Q\cap E_I$. The affine transformation $t_{q}w$ of the affine Weyl group $Q_I⋊W_I$ of the dual subsystem ${\mathrm{\Phi }}_I^{\vee }$ fixes the orthogonal projection $a_I$ of $a$ to $E_I$.

In any locally finite affine root-wall arrangement, the stabilizer of a point $a_I$ is generated by the reflections in walls through that point. To see this, choose a ball about $a_I$ meeting no wall not containing $a_I$. Inside it, the finitely many hyperplanes through $a_I$ cut local chambers; generic line segments join any two local chambers by crossing one wall at a time and avoiding their codimension-two intersections. The corresponding wall reflections therefore act transitively on the local chambers. If an affine Weyl element fixes $a_I$, choose a local chamber $C$ and a product $h$ of these reflections carrying $C$ to its image. Then $h^{-1}g$ preserves the global alcove containing $C$, so it is the identity by the simple transitivity in [F3]. Hence $g=h$. Apply this to $g=t_{q}w$ from [step 1.1] in the arrangement for the dual subsystem ${\mathrm{\Phi }}_I^{\vee }$. Every generating wall has a root $\alpha \in {\mathrm{\Phi }}_I$ and $⟨a,{\alpha }^{\vee }⟩\in \mathbb{Z}$; because $\alpha$ is in ${\mathrm{\Phi }}_I$, [F2] gives $⟨b,{\alpha }^{\vee }⟩=0$, so $⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}$. Taking finite parts shows that $w$ belongs to the subgroup $W_{\mathrm{i}\mathrm{n}\mathrm{t}}$ generated by the reflections $s_{\alpha }$ for roots with integral $\kappa$-pairing. Undo now the temporary Weyl conjugation of step 1.1: if that conjugating element was $u$, a generating root $\alpha$ in the conjugated coordinates returns to $u^{-1}\alpha$, and $⟨u\kappa ,{\alpha }^{\vee }⟩=⟨\kappa ,(u^{-1}\alpha {)}^{\vee }⟩$. Thus the original $w$ lies in the integral-root reflection subgroup for the original $\kappa$. From this point on $\kappa ,w,q,{\mathrm{\Phi }}^{+}$ and $Q_{+}$ refer to the original data. No preservation of antidominance or of the positive cone by $u$ is assumed.

The set ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\{\alpha \in \mathrm{\Phi }:⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}\}$ is stable under its root reflections: if $m=⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ and $\beta \in {\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$, then $⟨\kappa ,(s_{\alpha }\beta {)}^{\vee }⟩=⟨s_{\alpha }\kappa ,{\beta }^{\vee }⟩=⟨\kappa ,{\beta }^{\vee }⟩-m⟨\alpha ,{\beta }^{\vee }⟩\in \mathbb{Z},$ using crystallographic integrality. It is therefore a finite root system in its span, with positive system ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}\cap {\mathrm{\Phi }}^{+}$ and its own simple roots ${\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$. For every $\delta \in {\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$, antidominance makes the integer $⟨\kappa ,{\delta }^{\vee }⟩$ nonpositive. If $w=s_{{\delta }_1}\cdots s_{{\delta }_m}$ is reduced in $W_{\mathrm{i}\mathrm{n}\mathrm{t}}$, telescoping gives $\kappa -w\kappa ={\sum }_{j=1}^m⟨\kappa ,{\delta }_j^{\vee }⟩s_{{\delta }_1}\cdots s_{{\delta }_{j-1}}{\delta }_j\mathrm{.}$ Every prefix root is positive in ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ by [F4], hence is positive in the ambient system; every term lies in $-Q_{+}$. Thus $q\in -Q_{+}$. If $\kappa$ is regular, every integral pairing here is nonzero, so every coefficient is strictly negative.

Combining [step 1.1] and [step 3.1], an assumed $q\in Q_{+}\setminus \{0\}$ also lies in $-Q_{+}$, contradicting [F1]. This proves the antidominant assertion. If $\kappa$ is regular, every integral root pairing in [step 3.1] is strictly negative. For any nonidentity $w\in W_{\mathrm{i}\mathrm{n}\mathrm{t}}$, its reduced word has at least one term, and each term is a strictly negative integer multiple of a positive root; the sum is nonzero because $Q_{+}$ is pointed. Hence $q=0$ forces $w=1$, proving the regular assertion.

For each point $x\in X$, choose an embedding $M_x\hookrightarrow I_x$ into an injective ${\mathcal{A}}_x$-module by [F1]. The skyscraper sheaf $i_{x\ast }{I}_x$ has sections $I_x$ on opens containing $x$ and zero on opens not containing $x$, with $\mathcal{A}(U)$ acting through ${\mathcal{A}}_x$. The natural bijection ${\mathrm{Hom}}_{\mathcal{A}}(N,i_{x\ast }{I}_x)\cong {\mathrm{Hom}}_{{\mathcal{A}}_x}(N_x,I_x)$ shows $i_{x\ast }{I}_x$ is injective, since taking stalks is exact by [F2]. The product of these injectives is injective: Hom into the product is the product of Hom functors, and products of exact sequences of abelian groups are exact under the stated Axiom of Choice. The diagonal map $M\to {\prod }_{x\in X}{i}_{x\ast }{I}_x$ is injective on each stalk by construction. Thus $\mathrm{Mod}(\mathcal{A})$ has enough injectives.

Let $j:U\hookrightarrow X$ be open and $M$ an $\mathcal{A}{\mathrm{\mid }}_U$-module. Define $(j_{!}M)(V)$ as the sections of $M(V\cap U)$ whose support is closed in $V$, using Extension by zero for abelian sheaves on an open subspace. For $a\in \mathcal{A}(V)$ and such a section $s$, let $as$ be the action of $a{\mathrm{\mid }}_{V\cap U}$; if $s$ vanishes on $W\cap U$ near a point of $V\setminus U$, then so does $as$, so the support condition is preserved. The stalks are $M_x$ for $x\in U$ and zero otherwise by the underlying abelian construction. The abelian-sheaf adjunction extends a module map on $U$ uniquely to an abelian-sheaf map on $X$; the extension is $\mathcal{A}$-linear on stalks in $U$ by the original module-linearity and off $U$ because its source stalk is zero. Thus $j_{!}$ is left adjoint to restriction in module sheaves. The stalk description and [F2] show that it is exact.

Let $I$ be an injective $\mathcal{A}$-module sheaf and let $U\subseteq V$ be open. A section $s\in I(U)$ corresponds by evaluation at $1$ to a morphism $j_{!}{\mathcal{A}}_U\to I{\mathrm{\mid }}_V$. The natural map $j_{!}{\mathcal{A}}_U\to {\mathcal{A}}_V$ is a monomorphism: on stalks it is the identity for points of $U$ and the map $0\to {\mathcal{A}}_x$ outside $U$. Restriction preserves injectivity here because its exact left adjoint is extension by zero [F3, step 1.2]. Injectivity of $I{\mathrm{\mid }}_V$ extends the morphism across $j_{!}{\mathcal{A}}_U\hookrightarrow {\mathcal{A}}_V$. The extension is determined by the image $t$ of $1\in \mathcal{A}(V)$, and $t{\mathrm{\mid }}_U=s$. Every restriction $I(V)\to I(U)$ is therefore surjective, so $I$ is flasque.

Take an injective resolution $0\to M\to I^0\to I^1\to \cdots$ in $\mathrm{Mod}(\mathcal{A})$, which exists by step 1.1. Its global-sections complex computes module-derived global sections by definition. Each underlying abelian sheaf $I^p$ is flasque by step 2.1 and hence $\mathrm{\Gamma }$-acyclic by [F4]. Use the supplied abelian-sheaf datum of [F5], whose domain is all of $\mathrm{A}\mathrm{b}(X)$: the underlying $M$, every $I^p$, and every cycle are abelian sheaves and therefore belong to that domain. Forgetting the module action preserves exactness by [F2], so this is a $\mathrm{\Gamma }$-acyclic resolution relative to that datum. The acyclic-resolution theorem now identifies its global-sections cohomology with $H^q(X,M)$; its Dependent Choice hypothesis follows from the assumed AC. Comparison of resolutions makes the identification natural in $M$, and it preserves degree zero.

The Cech complex for $U_0,U_1$ has $C^0=\mathbb{C}[z]e_0\oplus \mathbb{C}[w]e_1,C^1=\mathbb{C}[z,z^{-1}]e_0,$ with differential $(f_0,f_1)\mapsto z^d{f}_1(1\mathrm{/}z)-f_0(z)$. Its kernel consists of polynomials $f_0(z)$ that also lie in $z^d\mathbb{C}[z^{-1}]$. If $d\ge 0$, these are exactly the span of $1,z,\dots ,z^d$; if $d<0$, the exponent ranges $\{0,1,\dots \}$ and $\{d,d-1,\dots \}$ are disjoint, so the kernel is zero. This proves the assertion for $H^0$.

The degree-one cohomology is the Laurent-polynomial quotient $\mathbb{C}[z,z^{-1}]\mathrm{/}(\mathbb{C}[z]+z^d\mathbb{C}[z^{-1}])\mathrm{.}$ The first summand spans exponents $n\ge 0$ and the second spans $n\le d$. If $d\ge -1$, these ranges cover every integer, so the quotient is zero. If $d\le -2$, the missing exponents are exactly $d+1,d+2,\dots ,-1$, which gives the stated basis and dimension $-d-1$. The Cech complex has no degree above one, so $H^q=0$ for $q\ge 2$.

On a principal ${\mathfrak{s}\mathfrak{l}}_2$ string with highest weight $2m>0$, $\mathrm{ad}h$ has weights $-2m,-2m+2,\dots ,2m$. For every $j\ge 1$, if weight $2j$ occurs then weight $2j-2$ occurs, and the raising operator $\mathrm{ad}e$ maps the latter one-dimensional weight space onto the former. Summing over the strings in [F1] gives $[e,{\mathfrak{n}}_j^{+}]={\mathfrak{n}}_{j+1}^{+}(j\ge 1),$ with zero spaces understood above the maximal root height.

Let $x\in {\mathfrak{n}}^{+}$ have nonzero simple coefficients. The map $T\to ({\mathbb{C}}^{\times }{)}^r$, $t\mapsto ({\alpha }_i(t){)}_i$, is an isogeny: its character-lattice map has image the full-rank root lattice. An isogeny of complex tori is surjective on complex points, so conjugation by one $t\in T$ sets all simple coefficients of $x$ to $1$. Now suppose heights below $j+1$ already agree with those of $e$. Their height-$(j+1)$ difference lies in ${\mathfrak{n}}_{j+1}^{+}$; by [step 1.1] choose $u_j\in {\mathfrak{n}}_j^{+}$ whose bracket with $e$ cancels that difference. The finite exponential $\exp (\mathrm{ad}{u}_j)$ changes no height at most $j$ and changes height $j+1$ by $[u_j,e]$; all its higher commutators have height at least $j+2$. Induction over the finitely many root heights conjugates $x$ to $e$ by $U$.

Conversely, conjugation by $T$ scales each simple-root coefficient by a nonzero scalar. For $u\in U$, every commutator in $\exp (\mathrm{ad}u)e-e$ has height at least two, because $u$ has positive height and $e$ has height one. Thus $U$ fixes the simple-root coefficients, and $B=TU$ preserves their nonvanishing. The inclusion in the right side and the reverse inclusion from step 2.1 prove the equality. The rank-zero case has ${\mathfrak{n}}^{+}=0$, $e=0$ and both sides are the singleton $\{0\}$.

Choose a very ample embedding $X\hookrightarrow {\mathbb{P}}^N$ and its standard affine cover $U_i=X\cap D_{+}(x_i)$. For a section $s\in \mathrm{\Gamma }(U_i,\mathcal{M})$, each overlap $U_i\cap U_j$ is the principal open $D(x_i\mathrm{/}{x}_j)$ in the affine $U_j$. By [F2] write $s{\mathrm{\mid }}_{U_i\cap U_j}=m_j\mathrm{/}(x_i\mathrm{/}{x}_j{)}^{r_j}$ for some $m_j\in \mathrm{\Gamma }(U_j,\mathcal{M})$. There are finitely many $j$, so choose a common $r\ge r_j$ and replace each numerator $m_j$ by $(x_i\mathrm{/}{x}_j{)}^{r-r_j}{m}_j$, so its denominator is now $(x_i\mathrm{/}{x}_j{)}^r$. With $x_j$ as the frame of $L={\mathcal{O}}_X(1)$ on $U_j$, the section $m_j\otimes x_j^r$ of $\mathcal{M}\otimes L^r$ extends $s\otimes x_i^r$ from $U_i\cap U_j$ to $U_j$. On an overlap $U_j\cap U_k$, the two extensions agree after inverting $x_i$, because both restrict to $s\otimes x_i^r$ there. Their difference is therefore killed by some power of $x_i$ in the quasi-coherent module of sections on that affine overlap. There are finitely many overlaps, so one further common power kills all differences; multiplying every local extension by that power of $x_i$ makes them agree and they glue to a global section of $\mathcal{M}\otimes L^n$ whose restriction to $U_i$ is $s\otimes x_i^n$. Thus each local section on a standard chart is locally in the image of a global map $L^{-n}\to \mathcal{M}$.

Take the direct sum of the maps $L^{-n}\to \mathcal{M}$ produced by [step 1.1] for all standard charts and all their local sections. At every stalk this map is surjective because the standard charts cover $X$ and their sections generate the associated modules. Hence $\mathcal{M}$ is the union of the images of finite subsums: the indexing set of finite subsums is filtered, their transition maps are inclusions, and their colimit is the image of the surjection. Each finite subsum is a finite direct sum of invertible sheaves, so its image is a finite-type quasi-coherent submodule of $\mathcal{M}$; it is coherent by [F3]. This proves the first assertion.

Fix the finite affine cover from [F1]. Every finite intersection is affine, and [F2] identifies sections on it with module sections; these sections commute with the filtered colimit. Therefore the ordered Cech complex of $\mathcal{M}$ is the degreewise filtered colimit of the ordered Cech complexes of ${\mathcal{F}}_j$. The cover computes sheaf cohomology by [F4], and exactness of filtered colimits in [F5] gives for every $q\ge 0$. The finite cover and degreewise finite sums are essential; no interchange with an infinite product is used.

Work on a Zariski open set where $T_X$ and ${\mathfrak{b}}_X$ are free and the anchor splits as an ${\mathcal{O}}_X$-module map. Combine tangent lifts $x_1,\dots ,x_d$ and a frame $b_1,\dots ,b_s$ of ${\mathfrak{b}}_X$ into one frame $e_1,\dots ,e_n$ of the full action algebroid, ordered with the tangent lifts first and the isotropy frame last. For every frame generator use the relations $e_{a}f=f{e}_a+a(e_a)(f)$ and, for $b>a$, $e_b{e}_a=e_a{e}_b+[e_b,e_a]$ to move functions left and order all generators; write each bracket in the chosen frame with its local function coefficients. Reordering terminates by lexicographic induction on generator degree, the number of generator/function inversions (a generator to the left of a coefficient), and the inversion count of the frame-generator word. A commutator correction lowers generator degree; the leading function move lowers the second count, and an out-of-order generator swap lowers the third. The only overlap types are $e_a(fg)$, $e_b{e}_{a}f$, and $e_c{e}_b{e}_a$ for arbitrary frame indices, including isotropy indices. Their reductions agree by the derivation rule for the anchor, the anchor identity $a([e_b,e_a])=[a(e_b),a(e_a)]$, and Jacobi for the action-algebroid bracket [F2]. A terminating rewriting system with every such overlap resolved has unique normal forms: induction on maximum remaining reduction length joins any two first reductions at their common overlap, after which induction identifies the normal forms. Hence the ordered monomials in all $e_a$, with coefficients on the left, form a local ${\mathcal{O}}_X$-basis, and the indicated ordering puts every tangent-lift monomial before isotropy monomials. Their leading symbols are the corresponding symmetric monomials, proving $\mathrm{gr}{\mathcal{U}}_X(\mathfrak{g})\cong {\mathrm{Sym}}_{{\mathcal{O}}_X}({\mathcal{O}}_X\otimes \mathfrak{g})$.

On a chart pulled back to $G$, the isotropy character is the constant $B$-character $\kappa +\rho$. It kills $[\mathfrak{b},\mathfrak{b}]$ and is invariant under $B$-conjugation; equivariance therefore gives ${\chi }^X([x,\xi ])=a(x)({\chi }^X(\xi ))$ for every local algebroid section $x$ and isotropy section $\xi$. In the ordered forms of step 1.1, use $b_a={\chi }^X(b_a)$ to eliminate each isotropy generator. The overlaps of this elimination with two isotropy generators agree because ${\chi }^X$ kills their bracket; the overlaps with a tangent lift agree by the displayed invariance identity; and overlaps with functions agree because $a(b_a)=0$. Thus the quotient has the ordered tangent-lift monomials as a local basis, and its associated graded is ${\mathrm{Sym}}_{{\mathcal{O}}_X}({\mathcal{O}}_X\otimes \mathfrak{g})\mathrm{/}({\mathfrak{b}}_X)\cong {\mathrm{Sym}}_{{\mathcal{O}}_X}{T}_X\mathrm{.}$ The degree-zero part is ${\mathcal{O}}_X$, and the first-order commutator is the anchor inherited from ${\mathcal{U}}_X(\mathfrak{g})$. This proves the TDO axioms and the two-sidedness of the defining ideal.

Suppose $\kappa +\rho$ is integral and put $L={\mathcal{O}}_X(\kappa +\rho )$. The infinitesimal G-linearization of $L$ gives a filtered algebra map ${\mathcal{U}}_X(\mathfrak{g})\to \mathrm{Diff}(L)$. Its isotropy action on each fiber is the character $\kappa +\rho$, so the map kills the ideal in step 2.1 and descends to ${\mathcal{D}}_{\kappa }\to \mathrm{Diff}(L)$. On associated graded it is the identity map $\mathrm{Sym}{T}_X\to \mathrm{Sym}{T}_X$ because both maps send a vector field to its principal symbol. A filtered map with associated-graded isomorphism is an isomorphism: lift any section order by order for surjectivity, and take the leading symbol of a kernel section for injectivity. Thus ${\mathcal{D}}_{\kappa }\cong \mathrm{Diff}(L)$. Taking $\kappa =-\rho$ gives $L={\mathcal{O}}_X$ and the asserted untwisted identity.

For integral $\eta$, write $L_{\eta }={\mathcal{O}}_X(\eta )$. Locally choose a frame $e$ of $L_{\eta }$. Tensoring a ${\mathcal{D}}_{\kappa }$-module with $e$ identifies its local differential-operator action with the action for the twist $\kappa +\eta$. If $e^{\mathrm{\prime }}=ue$ on an overlap, with $u$ an invertible regular function, the two local actions are related by conjugation $D\mapsto u^{-1}Du$; this is exactly the transition rule for the TDO parameter shifted by $\eta$. These local equivalences glue. Tensoring by $L_{-\eta }$ reverses the shift and composes with the first functor to the identity.

The orbit map $G\to N$, $g\mapsto \mathrm{Ad}(g)e$, has irreducible image closure. By [F1] its image has dimension $\dim \mathfrak{g}-\dim Z_{\mathfrak{g}}(e)=2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\dim N$; by [F6] the constructible image is dense and therefore contains a nonempty open subset of the irreducible variety $N$. The orbit is $G$-stable, so its translates of this open subset cover the orbit; consequently $G\cdot e$ is open in $N$.

The Springer map is $\mu :G{\times }^B{\mathfrak{n}}^{+}⟶N,[g,x]⟼\mathrm{Ad}(g)x\mathrm{.}$ Its graph is the incidence subvariety $\{(gB,y)\in X\times N:y\in \mathrm{Ad}(g){\mathfrak{n}}^{+}\},$ which is closed because the spaces $\mathrm{Ad}(g){\mathfrak{n}}^{+}$ form a vector subbundle of $X\times \mathfrak{g}$. The projection $X\times N\to N$ is projective by [F4], so $\mu$ is proper by the projective-morphism theorem included in [F4]. Surjectivity follows from the last clause of [F1].

If $y\in N\setminus G\cdot e$, conjugate it into ${\mathfrak{n}}^{+}$ by [F1]. At least one simple-root coefficient must vanish by [F2]. For each simple ${\alpha }_i$, set ${\mathfrak{n}}_i={⨁}_{\beta \in {\mathrm{\Phi }}^{+},\beta \mathrm{\ne }{\alpha }_i}{\mathfrak{g}}_{\beta }$. This is a nilpotent Lie subalgebra. It is an ideal in ${\mathfrak{p}}_{{\alpha }_i}=\mathfrak{b}\oplus {\mathfrak{g}}_{-{\alpha }_i}$: brackets with $\mathfrak{h}$ preserve each root space; brackets with positive root spaces have weights $\beta +\gamma$, which if nonzero remain positive and cannot equal ${\alpha }_i$ because $\beta \mathrm{\ne }{\alpha }_i$ has a positive coefficient at another simple root; and if $\beta -{\alpha }_i$ is a root for $\beta \mathrm{\ne }{\alpha }_i$, it must be positive, since otherwise ${\alpha }_i$ would be a sum of positive roots. These assertions use [F5] and the simple-root basis. The exponential/BCH coordinates in [F5] identify $\exp :{\mathfrak{n}}^{+}\to U$ as a polynomial isomorphism, so $N_i=\exp ({\mathfrak{n}}_i)$ is a closed connected unipotent subgroup with Lie algebra ${\mathfrak{n}}_i$. Order the positive-root coordinates with ${\alpha }_i$ first. The product of the remaining root subgroups is then a closed subvariety of $U$ of dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1$; it lies in $N_i$, which has the same dimension, so equals $N_i$. Hence $U=U_{{\alpha }_i}{N}_i$. Since ${\mathfrak{n}}_i$ is an ideal, its normalizer in the connected group $P_{{\alpha }_i}$ has full Lie algebra and hence equals $P_{{\alpha }_i}$; thus $N_i$ is normal in $P_{{\alpha }_i}$. The quotient $P_{{\alpha }_i}\mathrm{/}{N}_i$ is generated by $T$ and the images of $U_{{\alpha }_i}$ and $U_{-{\alpha }_i}$. The rank-one homomorphism of Rank-one SL2 homomorphism and Weyl representative supplies the semisimple rank-one subgroup. The central subtorus $(\ker {\alpha }_i{)}^0\subseteq T$ centralizes it, and ${\alpha }_i\circ {\alpha }_i^{\vee }(t)=t^2$ shows that its quotient in $T$ is covered by the coroot torus up to finite index. Hence $P_{{\alpha }_i}\mathrm{/}{N}_i$ is a central product of a torus and a semisimple rank-one group, so it is reductive and has no nontrivial connected normal unipotent subgroup. By Radical, unipotent radical, semisimple and reductive algebraic groups, $N_i$ is the unipotent radical of $P_{{\alpha }_i}$. It is therefore $P_{{\alpha }_i}$-stable and has dimension $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1$. Thus $N\setminus G\cdot e$ is contained in the finite union of the images of $G{\times }^{P_{{\alpha }_i}}{N}_i⟶N,[g,u]⟼\mathrm{Ad}(g)\log u\mathrm{.}$ Here $\log :N_i\to {\mathfrak{n}}_i$ is the polynomial inverse of the exponential just constructed; it is $P_{{\alpha }_i}$-equivariant by conjugation, so the displayed map is well-defined. Each source has dimension $(\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1)+(\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-1)=2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-2$ by [F4]. Hence the complement of the open regular orbit has codimension at least two. The Jacobian rank at $e$ is full by the proof of [F1]; $G$-equivariance makes every point of $G\cdot e$ regular. Thus $N$ is regular in codimension one. Since it is an integral complete intersection, [F7] gives normality.

The fiber over $e$ is a single point. Indeed, if $e\in \mathrm{Ad}(g){\mathfrak{n}}^{+}$, write $g=b_1{n}_w{b}_2$ by [F3]. The element $x=\mathrm{Ad}(b_1^{-1})e$ still has every simple coefficient nonzero by [F2]. Since $\mathrm{Ad}(n_w^{-1})x\in {\mathfrak{n}}^{+}$, each nonzero simple-root component of $x$ must map to a positive root space; distinct root spaces cannot cancel. Therefore $w^{-1}$ sends every positive simple root to a positive root, so $w=1$ by [F3], and $gB=B$. This algebraic argument holds after every algebraically closed extension of $\mathbb{C}$ as well. Equivariance gives a singleton geometric fiber over every point of $O=G\cdot e$. The restriction ${\mu }_O:{\mu }^{-1}(O)\to O$ is proper by base change. Its fibers are finite type and their geometric point sets are singletons, so each fiber point is isolated and has residue field finite over the base residue field; the finite-fiber criterion in [F8] makes ${\mu }_O$ quasi-finite, and proper quasi-finite finiteness in [F8] makes it finite. Both source and target are integral of dimension $2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, so the induced function-field extension is finite. It is separable in characteristic zero, and by [F9] has a primitive element $\theta$. After replacing an affine open $\mathrm{Spec}A\subseteq O$ by a smaller nonempty principal open, its finite preimage is $\mathrm{Spec}B$ with $B$ finite over $A$, $\theta$ lies in $B$, and $B=A[\theta ]\cong A[T]\mathrm{/}(f)$ for the monic minimal polynomial $f$ of $\theta$: first clear denominators to place $\theta$ in $B$, then localize to kill the finite torsion quotient $B\mathrm{/}A[\theta ]$ and to put the coefficients of $f$ in $A$. The discriminant of $f$ is nonzero by separability, so after inverting it this finite map is étale of rank $d=[\mathrm{Frac}B:\mathrm{Frac}A]$. Every geometric closed fiber in this open then has $d$ distinct points, but every geometric fiber over $O$ is a singleton; therefore $d=1$. This proves birationality without inferring degree from a closed-point count alone.