The nilpotent central fiber is reduced and integral

If $\mathfrak{g}=0$ then $r=0$ and $A=\mathbb{C}$, so all assertions hold; assume otherwise. By [F1], $S(\mathfrak{g})$ is free over the polynomial invariant algebra $\mathbb{C}[p_1,\dots ,p_r]$. After quotienting by the first $j$ generators it remains free over $\mathbb{C}[p_{j+1},\dots ,p_r]$; hence multiplication by the next generator is injective and all successive quotients are nonzero. Thus the $p_i$ form a regular sequence, rather than merely a sequence of algebraically independent elements. The Koszul Euler characteristic gives $\mathrm{Hilb}(A,t)={\prod }_i(1-t^{d_i})\mathrm{/}(1-t{)}^{\dim \mathfrak{g}}$. At any rational maximal ideal containing the $p_i$, the ambient polynomial local ring has dimension $\dim \mathfrak{g}$: the chain of translated coordinate ideals gives this lower bound at a rational point, and A polynomial ring in n variables over a field has dimension n gives the upper bound, and [F2] gives quotient dimension $\dim \mathfrak{g}-r$. At every prime the same regular sequence localizes and [F2] makes the quotient Cohen--Macaulay. In particular $\dim A=\dim \mathfrak{g}-r=2N$ by root-space decomposition. The Hilbert series has pole order $2N$ at $t=1$; it is not a polynomial of degree $2N$.

Choose for every simple root ${\alpha }_i$ an ${\mathfrak{s}\mathfrak{l}}_2$-triple $(e_i,f_i,h_i)$ and write $2{\rho }^{\vee }={\sum }_i{c}_i{h}_i$ in the simple-coroot basis, so ${\sum }_i{c}_i{\alpha }_j(h_i)=2$ for every simple root ${\alpha }_j$, and put $e={\sum }_i{e}_i$, $h=2{\rho }^{\vee }$, $f={\sum }_i{c}_i{f}_i$. Distinct simple-root differences are not roots, so $[e_i,f_j]=0$ for $i\mathrm{\ne }j$, and consequently $[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=h$. The eigenvalues of $\mathrm{ad}h$ are $2\mathrm{ht}(\alpha )$ and zero, hence all are even. Complete reducibility for this ${\mathfrak{s}\mathfrak{l}}_2$ subalgebra writes the adjoint module as a direct sum of ${\mathfrak{s}\mathfrak{l}}_2$-strings $L(2m_1)\oplus \cdots$; each string contributes exactly one vector killed by $h$, and $\ker \mathrm{ad}h=\mathfrak{h}$ has dimension $r$, so there are exactly $r$ strings; a string with $m_j=0$ would consist of a nonzero vector killed by $e$ and $h$, i.e. a nonzero element of $\mathfrak{h}$ commuting with all $e_i$, which is central in $\mathfrak{g}$, hence zero; so all $m_j\ge 1$ and counting dimensions gives ${\sum }_j(2m_j+1)=\dim \mathfrak{g}=2N+r$, that is ${\sum }_j{m}_j=N$. The highest vector of a string has $\mathrm{ad}h$-eigenvalue $2m_j>0$, hence lies in ${\mathfrak{n}}^{+}={⨁}_{\alpha >0}{\mathfrak{g}}_{\alpha }$; therefore $\ker \mathrm{ad}e$ is the $r$-dimensional span of the highest vectors and lies in ${\mathfrak{n}}^{+}$, and the lowest vectors span $z_f:=\ker \mathrm{ad}f\subseteq {\mathfrak{n}}^{-}$ with $\mathfrak{g}=[\mathfrak{g},e]\oplus z_f$.

Let ${\phi }_t(x)=t\cdot \mathrm{Ad}(t^{-{\rho }^{\vee }})(x)$ for $t\in {\mathbb{C}}^{\times }$. Since $p_i$ is $\mathfrak{g}$-invariant, $p_i({\phi }_{t}x)=t^{d_i}{p}_i(x)$ for all $x\in \mathfrak{g}$; since $\mathrm{Ad}(t^{-{\rho }^{\vee }})f_i=t{f}_i$, the map ${\phi }_t$ preserves $z_f$, so with $q_i(y):=p_i(e+y)$ one has $q_i({\phi }_{t}y)=p_i({\phi }_t(e+y))=t^{d_i}{q}_i(y)$: each $q_i$ is weighted-homogeneous of weighted degree $d_i$ for the weights $\mathrm{deg}{y}_j=m_j+1$ on $z_f$; indeed a root vector of height $t$ has $\phi$-weight $1-t$, so the lowest vector $y_j$ has weight $m_j+1\ge 2$.

The group constructions needed below are explicit. Order the height spaces of $\mathfrak{g}$ by root height, with $\mathfrak{h}$ at height zero. The endomorphisms $\mathrm{ad}{\mathfrak{n}}^{+}$ strictly raise height. Exponential and logarithm are finite inverse polynomials on the corresponding strictly triangular matrices. For a unipotent automorphism $u$ which raises height relative to the identity, $\exp (t\log u)$ preserves the Lie bracket for every integer $t$ since it equals $u^t$; the bracket equations are polynomial in $t$, so they hold for all complex $t$. Differentiating at zero shows that $\log u$ is a derivation. By [F4] it equals $\mathrm{ad}x$; comparison of the height components shows $x\in {\mathfrak{n}}^{+}$, because negative-root components lower height and a Cartan component acts in height zero, and $\mathrm{ad}$ is injective. Thus $N_{+}=\exp (\mathrm{ad}{\mathfrak{n}}^{+})$ is exactly the closed subgroup of $G$ defined by these strict-height unipotence equations; it is polynomially isomorphic to ${\mathfrak{n}}^{+}$, hence connected. Let $H=({\mathbb{C}}^{\times }{)}^r$ act by fixing $\mathfrak{h}$ and multiplying ${\mathfrak{g}}_{\alpha }$ by ${\prod }_i{t}_i^{a_i}$ when $\alpha ={\sum }_i{a}_i{\alpha }_i$. Root brackets show this is a closed torus of automorphisms, with Lie algebra $\mathrm{ad}\mathfrak{h}$. It normalizes $N_{+}$; $B=H{N}_{+}$ is closed, since the diagonal entries determine its $H$ factor and the remaining strict-height unipotent equations determine its $N_{+}$ factor. It is connected and solvable, with Lie algebra $\mathrm{ad}\mathfrak{b}$. Any Lie subalgebra containing $\mathfrak{b}$ is $\mathfrak{h}$-stable and splits into root spaces; if it is larger it contains a negative root space and hence, together with its positive root space, a root ${\mathfrak{s}\mathfrak{l}}_2$, so it cannot be solvable. Thus $B$ is maximal among connected solvable closed subgroups (a larger such subgroup would have larger solvable Lie algebra), and is a Borel. The connected group $G=\mathrm{Aut}(\mathfrak{g}{)}^0$ is smooth and irreducible of dimension $\dim \mathfrak{g}$ by [F4], and its Borels are conjugate. These explicit tori also give the height scalings used in step 1.3.

The central reduction has associated graded $A$: fix a central character $\chi :Z\to \mathbb{C}$ with kernel ${\mathfrak{m}}_{\chi }$, and let $J={\mathfrak{m}}_{\chi }U(\mathfrak{g})$. By [F6], $\mathrm{gr}U(\mathfrak{g})=S(\mathfrak{g})$, $\mathrm{gr}Z=S(\mathfrak{g}{)}^{\mathfrak{g}}$, and the leading symbol of any $z\in {\mathfrak{m}}_{\chi }$ lies in $S(\mathfrak{g}{)}_{+}^{\mathfrak{g}}=(p_1,\dots ,p_r)$; conversely each $p_i$ is the leading symbol of an element of ${\mathfrak{m}}_{\chi }$, for instance of $z_i-\chi (z_i)$ for a central element $z_i$ of leading symbol $p_i$ (the scalar term has lower filtration degree because $d_i>0$). Therefore $\mathrm{gr}J=(p_1,\dots ,p_r)S(\mathfrak{g})$ and the natural graded surjection $S(\mathfrak{g})\mathrm{/}(p_1,\dots ,p_r)\twoheadrightarrow \mathrm{gr}(U(\mathfrak{g})\mathrm{/}J)$ is an isomorphism: both sides have the Hilbert series of step 1.1, since the filtered free harmonic decomposition of [F6] restricts to a basis of $U(\mathfrak{g})\mathrm{/}J$ whose symbols have exactly the Hilbert series ${\prod }_i(1-t^{d_i})\mathrm{/}(1-t{)}^{\dim \mathfrak{g}}$. Hence the PBW associated graded of $U(\mathfrak{g})\mathrm{/}U(\mathfrak{g})\ker \chi$ is canonically $A$.

The morphism $\mathrm{\Psi }:G\times z_f\to \mathfrak{g}$, $(g,y)\mapsto g\cdot (e+y)$, is dominant. Its differential at $(1,0)$ is $(\xi ,y)\mapsto [\xi ,e]+y$ for $\xi \in T_{1}G=\mathfrak{g}$ (using $G$-equivariance of the action), with image $[\mathfrak{g},e]+z_f=\mathfrak{g}$ by step 1.2; hence the cotangent map is injective and, by the submersion criterion at the rational points $(1,0)$ and $e$, $\mathrm{\Psi }$ is locally standard smooth, hence flat, at $(1,0)$; the flat locus is open and flat morphisms of finite presentation are open, so the image of $\mathrm{\Psi }$ contains a nonempty open subset of $\mathfrak{g}$; since $\mathfrak{g}$ is irreducible, $\mathrm{\Psi }$ is dominant.

If $\mathrm{ad}x$ is nilpotent, $U_x=\{\exp (t\mathrm{ad}x):t\in \mathbb{C}\}$ is a closed connected unipotent subgroup of $G$: for $x\mathrm{\ne }0$ logarithm identifies it with the closed line $\mathbb{C}\mathrm{ad}x$; for $x=0$ it is trivial. Among connected solvable closed subgroups containing it choose one of maximal dimension; any strictly larger connected closed subgroup has larger dimension, so it is a Borel. Conjugacy from [F4] moves it into the $B$ constructed in step 2.1. Every unipotent element of that triangular group has diagonal entries all one, hence trivial $H$ factor, and lies in $N_{+}$. Its logarithm therefore belongs to $\mathrm{ad}{\mathfrak{n}}^{+}$ by step 2.1. Injectivity and equivariance of $\mathrm{ad}$ show that $x$ is conjugate into ${\mathfrak{n}}^{+}$. Conversely the invariant $p_i$ vanishes on ${\mathfrak{n}}^{+}$: the torus automorphism multiplying height $j$ by $t^j$ sends every $n\in {\mathfrak{n}}^{+}$ to zero as $t\to 0$, while invariance preserves $p_i(n)$. Thus $p_i(n)=p_i(0)=0$. Lie invariance implies $G$-invariance here: the differential of $g\mapsto p_i(gx)$ vanishes on the connected smooth group, and in characteristic zero a regular function with zero differential on an irreducible variety is constant.

The restrictions $q_1,\dots ,q_r\in \mathbb{C}[z_f]$ are algebraically independent: the pullback ${\mathrm{\Psi }}^{\ast }$ is injective because $\mathrm{\Psi }$ is dominant, and ${\mathrm{\Psi }}^{\ast }(p_i)=q_i$ by $G$-invariance of the $p_i$; if a nonzero polynomial $F$ satisfied $F(q_1,\dots ,q_r)=0$, then ${\mathrm{\Psi }}^{\ast }(F(p_1,\dots ,p_r))=0$, hence $F(p_1,\dots ,p_r)=0$ in $S(\mathfrak{g})$, contradicting the algebraic independence of $p_1,\dots ,p_r$ from [F1].

The common zero set $\mathcal{N}=\{x:p_i(x)=0\text{ for all }i\}$ equals $G\cdot {\mathfrak{n}}^{+}$: the inclusion $\supseteq$ is step 3.2, and conversely if all $p_i$ vanish then every invariant polynomial of positive degree vanishes, so the invariant traces $\mathrm{tr}((\mathrm{ad}x{)}^k)$, $k\ge 1$, vanish because each is invariant with zero constant term; Newton's identities express every coefficient of the characteristic polynomial of $\mathrm{ad}x$ polynomially in these traces, so the characteristic polynomial is $T^{\dim \mathfrak{g}}$, hence $\mathrm{ad}x$ is nilpotent and $x$ is conjugate into ${\mathfrak{n}}^{+}$ by step 3.2. Also $\mathcal{N}$ is the underlying set of $\mathrm{Spec}A$.

The map $q=(q_1,\dots ,q_r):{\mathbb{A}}^r\to {\mathbb{A}}^r$ is dominant by step 4.1, and ${\mathbb{A}}^r$ is smooth, so generic smoothness provides a dense open subset of ${\mathbb{A}}^r$ on which the differential of $q$ is surjective; in particular the Jacobian determinant $D=\det (\mathrm{\partial }{q}_i\mathrm{/}\mathrm{\partial }{y}_j)$ is not identically zero.

Weights: $D$ is weighted-homogeneous of weighted degree ${\sum }_i{d}_i-{\sum }_j(m_j+1)={\sum }_i(d_i-1)-{\sum }_j{m}_j=N-N=0$ by steps 1.2 and 1.3, because the derivative with respect to $y_j$ lowers weighted degree by $m_j+1$; a nonzero polynomial all of whose monomials have positive weights $m_j+1$ and total weighted degree zero is a nonzero constant. Hence $D$ is a nonzero constant, and in particular the linear map $z_f\to {\mathbb{C}}^r$, $y\mapsto (d{p}_1(e)(y),\dots ,d{p}_r(e)(y))$, given by the matrix $(\mathrm{\partial }{q}_i\mathrm{/}\mathrm{\partial }{y}_j)(0)$, is invertible.

The Jacobian matrix of the presentation $A=S(\mathfrak{g})\mathrm{/}(p_1,\dots ,p_r)$ at the rational maximal ideal $\mathfrak{m}$ of $e$ has rank exactly $r$: its rows are the differentials $d{p}_i(e)$, which are linearly independent by step 6.1. Since $\dim A_{\mathfrak{m}}=\dim A=\dim \mathfrak{g}-r$ by step 1.1, the Jacobian criterion in the form $\mathrm{rank}J(\mathfrak{m})=n-\dim A_{\mathfrak{m}}$ identifies $A_{\mathfrak{m}}$ as a regular local ring.

The set $\mathcal{N}$ is irreducible: step 4.2 identifies its closed points with the image of the irreducible variety $G\times {\mathfrak{n}}^{+}$, so its closure is irreducible and equals the underlying space of $\mathrm{Spec}A$ (In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum and Over an algebraically closed field, every maximal ideal is an evaluation ideal give density of its complex rational closed points). In particular there is a unique minimal prime ${\mathfrak{p}}_0$. The regular local ring $A_{{\mathfrak{m}}_e}$ of step 7.1 localizes further to $A_{{\mathfrak{p}}_0}$; localizations of regular local rings are regular by localisations of regular local rings are regular. Since ${\mathfrak{p}}_0$ is minimal, that localization has dimension zero and is a field. This proves generic reducedness without needing an assertion about openness of the regular orbit.

There are no embedded associated primes. For $\mathfrak{q}\in \mathrm{Ass}(A)$ choose a maximal ideal $\mathfrak{m}\supseteq \mathfrak{q}$. An element with annihilator $\mathfrak{q}$ remains nonzero at $\mathfrak{m}$ and has annihilator $\mathfrak{q}{A}_{\mathfrak{m}}$, so this is an associated prime of $A_{\mathfrak{m}}$. That local ring is Cohen--Macaulay by step 1.1, and the associated-prime theorem in [F2] makes $\mathfrak{q}{A}_{\mathfrak{m}}$ minimal. Any strict smaller prime of $A$ below $\mathfrak{q}$ would remain strict after localization at $\mathfrak{m}$, so $\mathfrak{q}$ itself is minimal. Hence the only possible associated prime is the unique ${\mathfrak{p}}_0$ of step 8.1. The associated-localization detection map in [F5] therefore injects $A$ into the field $A_{{\mathfrak{p}}_0}$, proving that $A$ is an integral domain and in particular reduced. The Cohen--Macaulay argument is essential here; generic reducedness alone would not exclude embedded nilpotents.

Combining steps 1.1, 2.2, 9.1: $A$ is a reduced integral domain of dimension $\dim \mathfrak{g}-r=2N=2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ with the displayed Hilbert series, and every central reduction has associated graded $A$. The Axiom of Choice is inherited from the invariant-theory, group-theoretic and geometric suppliers [F1], [F4], [F5] and [F6]; the conclusions themselves are choice-free statements about $\mathfrak{g}$.