Normality of the nilpotent cone and Springer functions

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.

Let $R=\mathbb{C}[N]$ and $S=\mathrm{\Gamma }(T^{\ast }X,\mathcal{O})$. Properness makes $S=\mathrm{\Gamma }(N,{\mu }_{\ast }{\mathcal{O}}_{T^{\ast }X})$ a finite $R$-algebra by [F10]. Birationality embeds $S$ in $\mathrm{Frac}(R)$ and contains $R$. Every element of a finite $R$-algebra is integral over $R$ by the determinant trick. Since $R$ is integrally closed by step 2.1, $S=R$. This proves the Springer-functions isomorphism without inferring regularity of a rational function from connected fibers.