---
title: Subregular Nilpotents in Stable Gradings
url: https://www.emergentmind.com/topics/subregular-nilpotents-in-stable-gradings
type: topic
---

# Subregular Nilpotents in Stable Gradings

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Subregular nilpotents in stable gradings arise naturally in Vinberg’s $\theta$-group representations and give rise to uniform families of algebraic curves. In the setting of a $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebra $\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$ coming from a finite-order automorphism, the relevant degree-one representation is $V=\mathfrak h_1$ for the connected fixed-point group $G=(H^\theta)^\circ$. A nilpotent element $e\in \mathfrak h_1$ is called $\theta$-subregular when it is subregular in $\mathfrak h$ and satisfies $\dim \mathfrak h_0(e)=1$; this condition is precisely what makes the graded Slodowy slice $X_e$ into a family of curves over the Vinberg quotient $B=V/G$. Recent work classifies the subregular-adapted stable gradings for all simple $H$ and $m\geq 2$, while the broader theory of positive-rank stable gradings supplies the Weyl-group criterion, little Weyl groups, and Kostant-section framework in which these constructions sit [2508.09607] [1307.5765].

## 1. Stable gradings and Vinberg representations

Let $k$ be a field of characteristic zero, let $H/k$ be a reductive algebraic group, and let $\theta:\mu_m\to \mathrm{Aut}_H$ be a $\mu_m$-action. After choosing a primitive $m$-th root of unity $\zeta$, differentiation gives an eigenspace decomposition
$$
\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}} \mathfrak h_i,\qquad 
\mathfrak h_i=\{x\in \mathfrak h:\theta(\zeta)(x)=\zeta^i x\}.
$$
The associated Vinberg representation is the action of $G=(H^\theta)^\circ$ on $V:=\mathfrak h_1$ by restriction of the adjoint representation. A vector $v\in V$ is stable if its $G$-orbit is closed and the stabilizer $Z_G(v)$ is a finite group scheme; the grading is stable if $V_{\bar k}$ contains stable vectors [2508.09607].

For simple $H$ over an algebraically closed field, stability is characterized by the Weyl-group condition that $\theta$ be principal of order $m$ and that the coset $W\vartheta\subset \mathrm{Aut}(\Lambda,\Phi)$ contain an elliptic $\mathbb{Z}$-regular element of order $m$. In the language of positive-rank gradings, a Cartan subspace $c\subset g(1)$ is a maximal abelian subspace consisting of semisimple elements, its dimension is the rank of the grading, and the little Weyl group
$$
W(c)=N_{G_0}(c)/Z_{G_0}(c)
$$
acts faithfully on $c$. Chevalley restriction takes the form $k[g(1)]^{G_0}\simeq k[c]^{W(c)}$, so the invariant ring is polynomial; in the Vinberg setting this is compatible with Panyushev’s congruence rule, which selects the $H$-invariants $F_i$ whose restrictions generate $k[V]^G$ by the condition $\deg(F_i)+e_i\equiv 0\pmod m$ [1307.5765].

Stable gradings have two consequences used throughout the theory. First, in $V$ one has “stable = regular semisimple,” equivalently $x\in V$ is stable if and only if the discriminant $\Delta(x)\neq 0$. Second, the quotient $B=\mathrm{Spec}\,k[V]^G$ is coregular. These properties make the degree-one piece of a stable grading simultaneously amenable to GIT, invariant theory, and singularity-theoretic slicing [2508.09607].

## 2. Subregular nilpotents in degree one

In a simple Lie algebra $\mathfrak h$ over an algebraically closed field, a nilpotent element $e$ is subregular if $\dim \mathfrak h_e=\mathrm{rk}(H)+2$; equivalently, its adjoint orbit is the unique maximal non-regular nilpotent orbit. In the graded setting, the additional condition
$$
e\in \mathfrak h_1,\qquad \dim \mathfrak h_0(e)=1
$$
defines $\theta$-subregularity. The second condition is not decorative: it ensures that the graded transverse slice has relative dimension one, so the quotient map produces a family of curves rather than a higher-dimensional family [2508.09607].

Stable gradings always contain a regular nilpotent element in degree one. For principal gradings, the sum of simple root vectors $E=E_1+\cdots+E_\ell$ lies in $g(1)$ and is regular nilpotent. The position of subregular nilpotents is subtler. In the classification of stable gradings of positive rank, stable gradings with normalized Kac coordinate $s_0=1$ are attached, in exceptional types, to distinguished nilpotent classes $A$ recorded by Bala–Carter labels. Among these labels are the subregular classes $G_2(a1)$, $F_4(a1)$, $E_6(a1)$, $E_7(a1)$, and $E_8(a1)$, which identify stable gradings of subregular type [1307.5765].

A frequent source of confusion is the distinction between a grading being attached to a subregular distinguished nilpotent and a paper explicitly producing a representative of the subregular orbit inside $g(1)$. The positive-rank classification gives the former correspondence via Kac diagrams and $\theta_A$, but does not explicitly claim that a representative of the subregular orbit lies in $g(1)$. The later theory of subregular-adapted stable gradings does exactly that: it isolates the cases where a $\theta$-subregular element exists in degree one and then uses it to construct the family $X_e\to B$ [1307.5765] [2508.09607].

## 3. Graded Slodowy slices and the curve construction

Given a nilpotent element $e\in \mathfrak h_1$, choose a normal $\mathfrak{sl}_2$-triple $(e,h,f)$ with $h\in \mathfrak h_0$ and $f\in \mathfrak h_{-1}$. The affine Slodowy slice and its graded intersection are
$$
S_e:=e+\mathfrak h(f)\subset \mathfrak h,\qquad 
X_e:=S_e\cap V=e+\mathfrak h_1(f)\subset V.
$$
Restricting the Vinberg quotient $\pi:V\to B$ gives
$$
\phi:X_e\to B.
$$
The map $\phi$ is flat, and the multiplication map $G\times X_e\to V$ is smooth. If $e$ is $\theta$-subregular, then the fibers of $\phi$ are $1$-dimensional; moreover, under a mild “goodness” condition that holds for the $\theta$-subregular elements in the classification, the central fiber $\phi^{-1}(0)$ is reduced, connected, and has a unique singular point at $e$ [2508.09607].

The mechanism is controlled by a compatible $\mathbb G_m\times \mu_m$-action derived from the $\mathfrak{sl}_2$-triple and the grading:
$$
\rho(t)=t^2\cdot \mathrm{Ad}(\lambda(t^{-1})),\qquad 
\sigma(\zeta)=\zeta\cdot \theta(\zeta^{-1}),
$$
with $X_e=S_e^\sigma$. In coordinates $U_1\simeq \mathbb A^3$ whose weights are determined by $\rho$, the central fiber of $S_e$ is given by one of the simple surface singularities
$$
z^2 \pm y^3 + x^n = 0,
$$
possibly with an explicit $\Gamma$-action in the non-simply laced cases. Passing to $\sigma$-fixed points sets one of the coordinates $x,y,z$ to zero and yields the explicit plane or weighted-plane curve equations appearing in the classification [2508.09607].

The dimension formula explains why subregularity is the correct threshold. For stable $\theta$, one has $\dim V=\dim \mathfrak h_0+\dim B$. The nilpotent cone in $V$ has dimension $\dim G$, and nilpotent orbits have codimension equal to $\dim \mathfrak h_0(e)$. Thus the condition $\dim \mathfrak h_0(e)=1$ forces $\phi:X_e\to B$ to have relative dimension $1$, which is the precise geometric condition for a family of curves [2508.09607].

## 4. Classical subregular-adapted stable gradings

The classification of subregular-adapted stable gradings for simple $H$ and $m\geq 2$ is given up to isogeny of $G$ and canonical identification of $V$. In the classical series, the resulting families already display the main geometric forms: genus-$0$ degenerations, hyperelliptic families, and mixed hyperelliptic-trigonal families [2508.09607].

**Type $A_r$** occurs only in the Coxeter grading $m=r+1$. Here $G\simeq \mathbb G_m^r$, $V$ is a sum of characters, and the family is
$$
xy=p_{r+1}.
$$
This is a genus-$0$ family.

**Type ${}^2A_r$** with $m=2$ has $G\simeq \mathrm{PSO}_{r+1}$ acting on $V\simeq \mathrm{Sym}^2_0(r+1)$. The corresponding curves are hyperelliptic:
$$
y^2=x^{r+1}+p_2x^{r-1}+\cdots+p_{r+1}.
$$
The generators $p_d$ have degrees $d=2,3,\dots,r+1$.

**Type $B_r$** with $m=2$ has $G\simeq \mathrm{SO}_{r+1}\times \mathrm{SO}_r$ acting on $V\simeq \mathrm{Hom}((r+1),(r))$, and the family is
$$
y^2=x^{2r}+p_2x^{2r-2}+\cdots+p_{2r}.
$$
These are hyperelliptic curves, with invariant degrees $2,4,\dots,2r$.

**Type $C_r$** with $m=2$ has $G\simeq \mathrm{GL}_r$ on $V\simeq \mathrm{Sym}^2(r)\oplus \mathrm{Sym}^2(r)^\vee$. There are two families, depending on the subregular element:
$$
xy^2=x^r+p_2x^{r-1}+p_4x^{r-2}+\cdots+p_{2r},
$$
and
$$
y^2=x^r+p_2x^{r-1}+\cdots+p_{2r}.
$$
The classification identifies these as hyperelliptic or trigonal according to the weights entering the Slodowy construction.

**Type $D_r$** with $m=2$ splits according to parity and outer twisting. For even $r$, one has $G\simeq \mathrm{SO}_r\times \mathrm{SO}_r$ acting on $V\simeq (r)\boxtimes (r)$, with family
$$
y(xy+p'_r)=x^{r-1}+p_2x^{r-2}+\cdots+p_{2r-2}.
$$
There are two degree-$r$ invariants $p_r,p'_r$, reflecting the spinor splitting. For ${}^2D_r$ with $r$ odd, the analogous family is
$$
y(xy+p_r)=\cdots.
$$

In all of these classical cases, the invariants $p_d$ generate $k[V]^G$, and the discriminant $\Delta$ controls smoothness: the fiber over $b\in B$ is smooth when $\Delta(b)\neq 0$ [2508.09607].

## 5. Exceptional types and explicit families

The exceptional types exhibit the full range of phenomena emphasized by the classification: multiple stable orders $m$, non-simply laced cases, and several genus-one or elliptic families that already occur in arithmetic statistics. The same construction applies uniformly, but the resulting equations depend sharply on the grading order and on the $\theta$-weights of the invariant generators [2508.09607].

For **type $G_2$**, there are two relevant orders. When $m=2$, $G\simeq \mathrm{SL}_2\times \mathrm{SL}_2$ with $V\simeq (2)\boxtimes \mathrm{Sym}^3(2)$, and the families are
$$
y^2x=x^3+p_2x^2+p_6
\qquad\text{or}\qquad
y^2=x^3+p_2x^2+p_6.
$$
When $m=3$, in the Coxeter-twisted case, $G\simeq \mathrm{GL}_2$ with
$$
V\simeq (\mathrm{Sym}^3(2)\otimes \det^{-2})\oplus \det,
$$
and the family is the elliptic curve
$$
y^2=x^3+p_6.
$$

For **type $F_4$**, the case $m=2$ has $G\simeq \mathrm{Sp}_6\times \mathrm{SL}_2$ and $V\simeq \wedge^3_0(6)\boxtimes (2)$. Two families occur:
$$
y^3=x^4+(p_2x^2+p_8)y+(p_6x^2+p_{12}),
$$
and
$$
y^2=x^3+p_8x+p_{12}.
$$
For $m=3$, with $G\simeq \mathrm{SL}_2\times \mathrm{SL}_3$ and $V\simeq (2)\boxtimes \mathrm{Sym}^2(3)\oplus (2)\boxtimes 1$, the family is
$$
y^2=x^4+p_6x^2+p_{12}.
$$

For **type $E_6$**, three stable orders contribute. In the outer case ${}^2E_6$ with $m=2$, $G\simeq \mathrm{PSp}_8$ and $V\simeq \wedge^4_0(8)$, giving
$$
y^3=x^4+(p_2x^2+p_5x+p_8)y+(p_6x^2+p_9x+p_{12}).
$$
For $m=3$, $G\simeq \mathrm{SL}_3^3$ and $V\simeq (3)\boxtimes (3)\boxtimes (3)$, with family
$$
y^2=x^4+p_6x^2+p_9x+p_{12}.
$$
For the outer case ${}^2E_6$ with $m=4$, $G\simeq \mathrm{SL}_2\times \mathrm{SL}_4$ and $V\simeq (2)\boxtimes \mathrm{Sym}^2(4)$, yielding
$$
y^2=x^3+p_8x+p_{12}.
$$

For **type $E_7$** with $m=2$, $G\simeq \mathrm{SL}_8$ acts on $V\simeq \wedge^4(8)$, and the family is
$$
y^3=x^3y+p_{10}x^2+x(p_2y^2+p_8y+p_{14})+p_6y^2+p_{12}y+p_{18}.
$$

For **type $E_8$**, the classification yields three orders. When $m=2$, $G\simeq \mathrm{Spin}_{16}$ on the half-spin representation, with family
$$
y^3=x^5+(p_2x^3+p_8x^2+p_{14}x+p_{20})y+(p_{12}x^3+p_{18}x^2+p_{24}x+p_{30}).
$$
When $m=3$, $G\simeq \mathrm{SL}_9/\mu_3$ and $V\simeq \wedge^3(9)$, with family
$$
y^2=x^5+p_{12}x^3+p_{18}x^2+p_{24}x+p_{30}.
$$
When $m=5$, $G\simeq (\mathrm{SL}_5\times \mathrm{SL}_5)/\mu$ for $\mu=\{(\zeta,\zeta^2):\zeta\in \mu_5\}$ and $V\simeq (5)\boxtimes \wedge^2(5)$, with family
$$
y^2=x^3+p_{20}x+p_{30}.
$$

These exceptional families include hyperelliptic, trigonal, plane quartic, genus-one, and elliptic cases. The classification identifies them as arising from a single Lie-theoretic mechanism rather than as isolated invariant-theoretic constructions [2508.09607].

## 6. Relation to the earlier classification and conceptual caveats

The classification of families of curves associated with subregular nilpotents generalizes the earlier stable-grading framework in two directions. First, it extends Thorne’s $\mu_2$-grading constructions for simply laced types $A$, $D$, and $E$ to all $m\geq 2$. Second, it includes non-simply laced types $B_r$, $C_r$, $F_4$, and $G_2$. New cases explicitly singled out by the classification include the $m=3$ families for $E_6$, $F_4$, and $E_8$, the $m=4$ families for ${}^2E_6$ and $F_4$, and the $m=5$ family for $E_8$ [2508.09607].

The broader theory of positive-rank gradings supplies the ambient classification of stable gradings, their little Weyl groups, and their invariant degrees. It proves that stable $\Leftrightarrow$ principal $\Leftrightarrow$ elliptic $\mathbb Z$-regular on the root system, identifies $W(c,\theta)=W^\theta$ in the stable principal cases, and establishes Kostant sections for inner exceptional types through a Levi reduction. In particular, for inner $E_6$, $E_7$, and $E_8$, there exists a $\theta$-stable Levi $L_\theta$ such that $\theta|_{\mathrm{Lie}(L_\theta)}$ is principal and the graded invariant theory reduces to the principal case in the Levi [1307.5765].

Two clarifications are essential. The first is that subregular-adapted is stricter than stable. While every stable $\mu_2$-grading is subregular-adapted, for $m\geq 3$ there are stable gradings that are not subregular-adapted; the listed examples are $\mu_4$- and $\mu_8$-gradings on $E_8$, and $\mu_8$ on $F_4$. The second is that non-subregular-adapted gradings may still yield curve families, but these are often genus $0$ or transverse slices inside families already arising from subregular-adapted gradings. This suggests that $\theta$-subregularity isolates the cases in which the curve family is both intrinsic to the grading and maximally uniform [2508.09607].

## 7. Orbit parametrizations and the $E_8$, $m=5$ case

A central consequence of the classification is that almost all classical coregular representations used in arithmetic statistics can be interpreted as Vinberg representations attached to stable gradings and subregular slices. Hyperelliptic curves with a marked Weierstrass point arise from type ${}^2A_r$, $m=2$, where $V\simeq \mathrm{Sym}^2_0(r+1)$ for $G\simeq \mathrm{PSO}_{r+1}$ and the invariant ring is generated by $p_2,\dots,p_{r+1}$. Genus-one descent examples fit the same pattern: $n=3$ comes from type ${}^3D_4$, $m=3$, with $G\simeq \mathrm{PGL}_3$ acting on ternary cubic forms and invariants $p_4,p_6$; $n=4$ comes from ${}^2E_6$, $m=4$, with invariants $p_8,p_{12}$; and $n=5$ comes from type $E_8$, $m=5$, with invariants of degrees $20$ and $30$. In each case, the dictionary is “invariants $\to$ coefficients in the curve equation,” while integral orbits with specified invariants correspond to arithmetic objects via
$$
\ker\!\big(H^1(k,Z_G(v))\to H^1(k,G)\big)
$$
[2508.09607].

The extended example is the principal $\mathbb Z/5\mathbb Z$-grading of split $E_8$ over $\mathbb Q$. Here
$$
G\simeq (\mathrm{SL}_5\times \mathrm{SL}_5)/\mu,\qquad 
\mu=\{(\zeta,\zeta^2):\zeta\in \mu_5\},
$$
and
$$
V\simeq (5)\boxtimes \wedge^2(5).
$$
The invariant ring is
$$
k[V]^G=k[I,J],
$$
where $I$ and $J$ have degrees $20$ and $30$. The discriminant satisfies
$$
\Delta=\lambda\cdot (4I^3+27J^2)^4
$$
for some $\lambda\in \mathbb Q^\times$, and the subregular slice produces the elliptic family
$$
y^2=x^3+Ix+J.
$$
This equation is obtained from the graded Slodowy surface
$$
-y^2+x^3+x(p_2 z^3+p_8 z^2+p_{14} z+p_{20})+(z^5+p_{12} z^3+p_{18} z^2+p_{24} z+p_{30})=0
$$
by taking $\sigma$-fixed points and setting $z=0$, so that only $p_{20}$ and $p_{30}$ survive on $B=V/G$ [2508.09607].

The associated orbit-parametrization theorem states that there exists $N\geq 1$ such that, for any field $k/\mathbb Q$ and any parameters $A,B\in k$ with $4A^3+27B^2\neq 0$, the elliptic curve
$$
E_{A,B}: y^2=x^3+Ax+B
$$
admits a natural injection, functorial in $k$,
$$
\eta_{A,B}: E_{A,B}(k)/5E_{A,B}(k)\hookrightarrow G(k)\backslash V_{A,B}(k),
$$
where
$$
V_{A,B}(k)=\{v\in V(k): I(v)=A,\ J(v)=B\}.
$$
Over $\mathbb Q$, this extends to an injection
$$
\widetilde{\eta}_{A,B}: \mathrm{Sel}_5(E_{A,B}/\mathbb Q)\hookrightarrow G(\mathbb Q)\backslash V_{A,B}(\mathbb Q).
$$
If $A,B\in \mathbb Z$, then all orbits in the image have representatives in $(1/N)\cdot V(\mathbb Z)$, so they are integral up to uniformly bounded denominators [2508.09607].

The Lie-theoretic proof combines a Kostant section $\kappa:B\to V$ coming from a regular nilpotent in $\mathfrak h_1$, a $\theta$-subregular slice $X_e\to B$, and an identification over $B^s=(\Delta\neq 0)$ of the finite group scheme $Z_G(\kappa)$ with the $5$-torsion $E[5]$ of the elliptic curve family. The comparison passes through the Picard groups of the associated elliptic surfaces, where $\mathrm{Pic}(S_b)$ identifies with the $E_8$ root lattice and $\mu_5$-coinvariants yield $\mathrm{Pic}(C_b)[5]\simeq E_b[5]$. Applying the Bhargava–Gross cohomological description of orbits then produces the injection from $E(k)/5E(k)$ and, over $\mathbb Q$, from $\mathrm{Sel}_5(E/\mathbb Q)$ [2508.09607].

Source: https://www.emergentmind.com/topics/subregular-nilpotents-in-stable-gradings