Canonical forms for coadjoint orbits of maximal unipotent subgroups: Surkov's Conjecture 3.8 for root systems beyond G_2 and F_4

The maximal unipotent subgroup $N=\exp(\mathfrak{n})$ of a simple algebraic group of type $\Phi$ acts on the dual $\mathfrak{n}^*$ of the nilradical of a Borel subalgebra, and Kirillov's orbit method makes the coadjoint orbits the central object in the representation theory of $N$. Surkov (Communications in Mathematics 34 (2026)) classified these orbits for $\Phi=F_4$, with the analogous statement for $G_2$, by exhibiting an explicit set $S\subset\mathfrak{n}^*$ of canonical forms — defined by a combinatorial rank condition on the support of a linear form relative to the lexicographic order on positive roots — that meets every coadjoint orbit exactly once, and posed the closing Conjecture 3.8: for an arbitrary root system, the same construction again yields a unique canonical form on every coadjoint orbit. The conjecture stands open for every root system other than $G_2$ and $F_4$, that is, for the classical types $A_n$, $B_n$, $C_n$, $D_n$ and the exceptional types $E_6$, $E_7$, $E_8$. The problem asks for a proof, or an explicit counterexample, for this canonical-forms cross-section conjecture.

Background

Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field of characteristic zero (or of sufficiently large characteristic), $\mathfrak{b}\subset\mathfrak{g}$ a Borel subalgebra (a maximal solvable subalgebra), and $\mathfrak{n}$ the nilradical of $\mathfrak{b}$, i.e. the span $\mathfrak{n}=\bigoplus_{\alpha\in\Phi^+}\mathbb{F} e_\alpha$ of the Chevalley root vectors $e_\alpha$ over the positive roots $\Phi^+$ of the root system $\Phi$. The group $N=\exp(\mathfrak{n})$ is the maximal unipotent subgroup of the corresponding simple algebraic group (equivalently, a Sylow $p$-subgroup of every Chevalley group of type $\Phi$ over $\mathbb{F}_q$), and it acts on the dual space $\mathfrak{n}^*$ by the coadjoint action, the transpose of the adjoint action: $(\exp(x)\cdot\lambda)(y)=\lambda(\exp(-\mathrm{ad}_x)(y))$ for $x,y\in\mathfrak{n}$, $\lambda\in\mathfrak{n}^*$. Under Kirillov's orbit method, coadjoint orbits parametrize the irreducible representations of a unipotent group, so their classification is the central problem of the subject — and a notoriously hard one: already for type $A_n$, where $N$ is the unitriangular group $UT(n+1)$ of upper-triangular matrices, a full classification of coadjoint orbits is known only for small matrix sizes, and the orbit counts are tied to the still-open Higman–Lehrer–Isaacs conjectures on the number of irreducible characters of unitriangular groups. Most orbits understood before 2025 were associated with rook placements: subsets $D\subset\Phi^+$ with $\alpha-\beta\notin\Phi^+$ for distinct $\alpha,\beta\in D$, equipped with nonzero scalars $\xi(\alpha)$; the corresponding separation results — forms sharing a rook-placement support but differing in scalars lie on distinct orbits — go back to André's theory of basic sums for $UT(n)$ (a stratification of $\mathfrak{n}^*$ by sums of coadjoint orbits indexed by combinatorial data) in type $A$, and to Ignatev–Surkov (2022) for $G_2$ and orthogonal placements in $F_4$. In a different direction, Goodwin–Mosch–Röhrle (2016) gave an algorithmic parametrization of all coadjoint orbits of $N$ via so-called minimal representatives for every simple type of rank at most $8$ except $E_8$, together with the number of orbits over finite fields; that parametrization is effective but is not a canonical-forms cross-section of the kind considered here. Surkov (2026) classified all coadjoint orbits for $\Phi=F_4$ as follows. Fix the total lexicographic order $\succ$ on $\Phi^+$ that compares simple-root coefficients $(a_1,a_2,\dots)$ in order. For $\lambda\in\mathfrak{n}^*$ define its support $\operatorname{supp}(\lambda)=\{\alpha\in\Phi^+:\lambda(e_\alpha)\neq 0\}$, and for $\gamma\in\Phi^+$ define the matrices $A_{\lambda,\gamma}=(\lambda([e_\alpha,e_\beta]))_{\alpha\in\Phi^+,\,\alpha\succeq\gamma,\,\beta\in\Phi^+}$ and $B_{\lambda,\gamma}=(\lambda([e_\alpha,e_\beta]))_{\alpha\in\Phi^+,\,\alpha\succ\gamma,\,\beta\in\Phi^+}$, so that $B_{\lambda,\gamma}$ is $A_{\lambda,\gamma}$ with the row numbered $\gamma$ deleted. Let $S\subset\mathfrak{n}^*$ be the set of $\lambda$ such that $\operatorname{rk}A_{\lambda,\gamma}=\operatorname{rk}B_{\lambda,\gamma}$ for every $\gamma\in\operatorname{supp}(\lambda)$; equivalently, for each root of the support, the row of $A_{\lambda,\gamma}$ numbered by that root is a linear combination of the rows indexed by larger roots in the order. Every rook-placement form automatically satisfies this rank condition, so $S$ extends the classical construction. Surkov's Theorem 1.1 proves that, over $\mathbb{F}_q$ of sufficiently large characteristic and over its algebraic closure, every coadjoint orbit of $N$ in type $F_4$ contains exactly one element of $S$: injectivity (distinct elements of $S$ lie on distinct orbits) is proved directly, and surjectivity is certified by computing $|S|$ over $\mathbb{F}_q$ and matching the orbit count of Goodwin–Mosch–Röhrle. The analogous statement for $G_2$ over $\mathbb{C}$ is established using the orbit equations of Ignatev–Surkov (2022). Immediately before the appendix, the paper states the natural generalization as Conjecture 3.8 and leaves it open: for an arbitrary root system $\Phi$ with the same construction, every coadjoint orbit should again contain a unique element of $S$. Whether this holds for any root system other than $G_2$ and $F_4$ is the open question.

Problem Statement

Let $\Phi$ be a root system other than $G_2$ and $F_4$ — a classical type $A_n$, $B_n$, $C_n$, or $D_n$, or an exceptional type $E_6$, $E_7$, or $E_8$ — realized with fixed positive roots $\Phi^+$ and Chevalley root vectors $e_\alpha$, $\alpha\in\Phi^+$, spanning the nilradical $\mathfrak{n}$ of a Borel subalgebra of the simple Lie algebra of type $\Phi$, over an algebraically closed field of characteristic $0$ or of sufficiently large characteristic (the source sets up its main theorem over finite fields $\mathbb{F}_q$ of sufficiently large characteristic and their algebraic closure, and treats the $G_2$ analogue over $\mathbb{C}$; characteristic $0$ is included as the natural limiting case of that convention). Fix the total lexicographic order $\succ$ on $\Phi^+$ obtained by comparing simple-root coefficients (with respect to the fixed numbering of the simple roots) in order: $\alpha=\sum_i a_i\alpha_i\succ\beta=\sum_i b_i\alpha_i$ if $a_j>b_j$ for the first index $j$ with $a_j\neq b_j$. For $\lambda\in\mathfrak{n}^*$ let $\operatorname{supp}(\lambda)=\{\alpha\in\Phi^+:\lambda(e_\alpha)\neq 0\}$, and for $\gamma\in\Phi^+$ let $A_{\lambda,\gamma}=(\lambda([e_\alpha,e_\beta]))_{\alpha\in\Phi^+,\,\alpha\succeq\gamma,\,\beta\in\Phi^+}$ and $B_{\lambda,\gamma}=(\lambda([e_\alpha,e_\beta]))_{\alpha\in\Phi^+,\,\alpha\succ\gamma,\,\beta\in\Phi^+}$, so that $B_{\lambda,\gamma}$ is $A_{\lambda,\gamma}$ with the row numbered $\gamma$ removed. Let $S\subset\mathfrak{n}^*$ be the set of $\lambda$ such that $\operatorname{rk}A_{\lambda,\gamma}=\operatorname{rk}B_{\lambda,\gamma}$ for every $\gamma\in\operatorname{supp}(\lambda)$. Prove or disprove Conjecture 3.8 of Surkov (2026) for the remaining types: for every coadjoint orbit of the maximal unipotent subgroup $N=\exp(\mathfrak{n})$ acting on $\mathfrak{n}^*$, there exists a unique linear form $\lambda\in S$ lying on that orbit. The problem is resolved by a proof of the conjecture for all remaining root systems, or by an explicit counterexample in any single remaining root system (two distinct elements of $S$ on one orbit, or an orbit containing no element of $S$).

References

1. Matvey A. Surkov, "Classification of coadjoint orbits for the maximal unipotent subgroup in the simple group of type F_4", Communications in Mathematics 34 (2026), no. 1, article 4. DOI: 10.46298/cm.16015; arXiv:2504.19701. https://doi.org/10.46298/cm.16015
2. Mikhail V. Ignatev, Matvey A. Surkov, "Rook placements in G_2 and F_4 and associated coadjoint orbits", Communications in Mathematics 30 (2022), no. 2, 129–149. DOI: 10.46298/cm.9041; arXiv:2107.03221. https://doi.org/10.46298/cm.9041
3. Simon M. Goodwin, Peter Mosch, Gerhard Röhrle, "On the coadjoint orbits of maximal unipotent subgroups of reductive groups", Transformation Groups 21 (2016), no. 2, 399–426. DOI: 10.1007/s00031-015-9318-9; arXiv:1409.4591. https://doi.org/10.1007/s00031-015-9318-9
4. Mikhail Ignatev, Alexey Petukhov, "Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types", Expositiones Mathematicae 44 (2026), no. 5, article 125795. DOI: 10.1016/j.exmath.2026.125795; arXiv:2507.20332. https://doi.org/10.1016/j.exmath.2026.125795
5. Mikhail Ignatev, Leonid Titov, "On the number of irreducible representations for finite unipotent Heisenberg-type groups", preprint (2026). arXiv:2604.01358. https://arxiv.org/abs/2604.01358
6. Mikhail Venchakov, "Rook placements and coadjoint orbits for maximal unipotent subgroups of finite symplectic groups", preprint (2026). arXiv:2602.14933. https://arxiv.org/abs/2602.14933
7. C. A. M. André, "Basic sums of coadjoint orbits of the unitriangular group", Journal of Algebra 176 (1995), no. 3, 959–1000. DOI: 10.1006/jabr.1995.1280. https://doi.org/10.1006/jabr.1995.1280