Classification of coadjoint orbits of maximal dimension for Borel nilradicals in types $B_n$, $D_n$ and $E_8$

The orbit method describes the irreducible representations of a maximal unipotent group through the coadjoint orbits of its Lie algebra, the nilradical $n$ of a Borel subalgebra of a simple Lie algebra $g$. The extremal case — classifying the coadjoint orbits of maximal possible dimension $2\mu(\Phi)$ — was solved for the upper-triangular (type $A$) case in Kirillov's 1962 founding paper, but a complete explicit description is still missing precisely for the orthogonal types $B_n$, $D_n$ and for the exceptional type $E_8$. Over finite fields of large characteristic the problem is equivalent to explicitly describing the irreducible complex characters of maximal degree $q^{\mu(\Phi)}$ of the maximal unipotent (Sylow $p$-) subgroups of the Chevalley groups of these types. For $B_n$ and $D_n$ the number of such orbits is already known while the orbits themselves are not; for $E_8$ even the count is unavailable. In the published literature the problem remains open: the type-$C$ case is resolved by André and Neto, all simple types of rank at most 8 except $E_8$ are covered algorithmically, and for $B_n$ and $D_n$ only an announced but still unpublished treatment by Ignatev and Venchakov exists.

Background

The orbit method, created by A. A. Kirillov in 1962, describes the representation theory of nilpotent Lie groups through the geometry of coadjoint orbits. Let $g$ be a finite-dimensional simple Lie algebra over an algebraically closed field $F$ of characteristic zero (or of sufficiently large characteristic), let $b$ be a Borel subalgebra of $g$ (a maximal solvable subalgebra), and let $n=[b,b]$ be the nilradical of $b$, i.e. its maximal nilpotent ideal. Concretely, if $\Phi$ is the root system of $g$ and $\Phi^+$ the positive roots determined by $b$, then $n$ is spanned by the root vectors $e_\alpha$ for $\alpha\in\Phi^+$. The unipotent group $N=\exp(n)$ acts on the dual space $n^*$; this dual action is called the coadjoint action and its orbits are the coadjoint orbits. The orbit method asserts a bijection between coadjoint orbits and irreducible representations: over $F=\mathbb C$, between coadjoint orbits and the continuous unitary irreducible representations of $N$ (as well as the primitive ideals of the universal enveloping algebra $U(n)$); over a finite field $\mathbb F_q$ of large enough characteristic, between coadjoint orbits and the complex irreducible characters of the finite group $N(q)$. In the finite-field case an orbit of dimension $2e$ corresponds to an irreducible character of degree exactly $q^e$, so orbit classification is equivalent to the character theory of $N(q)$, the maximal unipotent (Sylow $p$-) subgroup of a Chevalley group $G(q)$ (the group of $\mathbb F_q$-points of a split simple algebraic group).

A complete classification of all coadjoint orbits of $N$ on $n^*$ is a wild problem (in the representation-theoretic sense of containing the classification of pairs of matrices up to simultaneous conjugation, hence admitting no tractable parametrization), so one studies orbits of special dimensions. The extremal case — orbits of the maximal possible dimension $2\mu(\Phi)$, the maximum of the dimensions of all coadjoint $N$-orbits on $n^*$, equivalently the case where $q^{\mu(\Phi)}$ is the largest degree of an irreducible character of $N(q)$ — goes back to Kirillov's founding paper, which classified all maximal-dimensional orbits for the strictly upper-triangular matrices (type $A_{n-1}$). Much later, B. Kostant proved for a simple $g$ of any type that the coadjoint action of the full Borel group on $n^*$ has a unique Zariski-open orbit, that every $N$-orbit inside this open stratum has maximal possible dimension, and constructed the cascade of orthogonal roots — a canonical set of pairwise orthogonal positive roots built recursively from the simple roots — as an explicit cross-section for these open-stratum maximal orbits. Two positive roots are orthogonal if their Killing pairing vanishes; an orthogonal subset of $\Phi^+$ (in the finite-group literature also called an orthogonal rook placement, by analogy with non-attacking rooks on the root diagram) is a set of pairwise orthogonal positive roots, and to each such subset together with a tuple of nonzero scalars one attaches a linear form on $n^*$. Ignatyev proved that the orbit of such a form has a dimension computable from the Weyl group and constructed an explicit polarization for it — a subordinated subalgebra from which the orbit dimension is read off. Kostant's cascade, however, describes only the maximal orbits in the open stratum; maximal orbits lying in deeper strata require separate analysis.

The state of knowledge is type-dependent. Type $A_{n-1}$: complete since Kirillov (1962). Type $C_n$: a classification of the maximal-dimensional orbits follows from the supercharacter theory of André and Neto — a tractable, coarser substitute for the full character theory built from sums of characters indexed by certain subsets of positive roots — and Venchakov (2026) recently gave an independent classification together with an explicit formula for the maximal-degree characters. Types $B_n$, $D_n$: the number of maximal-dimensional orbits is known from the highest-degree character counts for the orthogonal Sylow subgroups in the tradition of Marjoram, but no explicit description of the orbits themselves has been published. Exceptional types: Goodwin, Mosch and Röhrle (2016) gave an algorithm that parametrizes all coadjoint orbits and ran it for every simple group of rank at most 8 except $E_8$, and Surkov (2026) published a complete explicit classification of all orbits for $F_4$; hence $E_8$ is the one simple type for which not even a computed parametrization exists. The 2025 paper of Ignatev and Petukhov classified all orbits of dimensions 2, 4 and 6 uniformly for all classical types and, as a corollary, proved for all classical Chevalley groups the conjecture of Isaacs that the number $O_e(q)$ of irreducible characters of degree $q^e$ of a maximal unipotent subgroup is a polynomial in $q-1$ with nonnegative integer coefficients, for exponents $e\le 3$; the same paper states explicitly that for $B_n$, $D_n$ and $E_8$ a complete description of orbits of maximal dimension is still unknown. This leaves the classification of maximal-dimensional coadjoint orbits for Borel nilradicals of types $B_n$, $D_n$ and $E_8$ as the surviving open core of this circle of problems.

Problem Statement

Let $g$ be a finite-dimensional simple Lie algebra of type $B_n$, $D_n$ or $E_8$ over an algebraically closed field of characteristic zero (or of sufficiently large characteristic), let $b\subset g$ be a Borel subalgebra with nilradical $n=[b,b]$, and let $N=\exp(n)$. Let $\Phi$ be the root system of $g$, and let $2\mu(\Phi)$ denote the maximal possible dimension of a coadjoint $N$-orbit on the dual space $n^*$, i.e. the maximum of the orbit dimensions; equivalently, over finite fields of sufficiently large characteristic, $q^{\mu(\Phi)}$ is the largest degree of an irreducible complex character of the finite group $N(q)$. Give a complete explicit classification of the coadjoint orbits of $N$ on $n^*$ having dimension $2\mu(\Phi)$: an explicit list of canonical forms, uniform in the rank and labeled by subsets of the positive roots together with finitely many scalar parameters, such that every orbit of dimension $2\mu(\Phi)$ contains exactly one form of the list — in the same sense in which Kirillov's 1962 classification describes the type-$A$ case. By the orbit method over finite fields of sufficiently large characteristic, an equivalent formulation is: describe explicitly all irreducible complex characters of maximal degree $q^{\mu(\Phi)}$ of the maximal unipotent subgroups $N(q)$ of the Chevalley groups of types $B_n$, $D_n$ and $E_8$. The classification should in particular produce, as a byproduct, the number of maximal-dimensional orbits for $E_8$, which is currently not known.

References

1. Mikhail Ignatev, Alexey Petukhov, Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types, arXiv:2507.20332 (2025), DOI 10.48550/arXiv.2507.20332, https://arxiv.org/abs/2507.20332
2. A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Mathematical Surveys 17 (1962), no. 4, 53-104, DOI 10.1070/RM1962v017n04ABEH004118, https://doi.org/10.1070/RM1962v017n04ABEH004118
3. Bertram Kostant, The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group, Moscow Mathematical Journal 12 (2012), no. 3, 605-620, DOI 10.17323/1609-4514-2012-12-3-605-620, https://arxiv.org/abs/1101.5382
4. Mikhail V. Ignatyev, Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups, Mathematical Notes 86 (2009), no. 1, 65-80, DOI 10.1134/S0001434609070074, https://arxiv.org/abs/0904.2841
5. Carlos A. M. André, Ana Margarida Neto, Super-characters of finite unipotent groups of types $B_n$, $C_n$ and $D_n$, Journal of Algebra 305 (2006), 394-429, DOI 10.1016/j.jalgebra.2006.04.030, https://doi.org/10.1016/j.jalgebra.2006.04.030
6. Simon M. Goodwin, Peter Mosch, Gerhard Röhrle, On the coadjoint orbits of maximal unipotent subgroups of reductive groups, Transformation Groups 21 (2016), 399-426, DOI 10.1007/s00031-015-9318-9, https://arxiv.org/abs/1409.4591
7. 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, DOI 10.46298/cm.16015, https://arxiv.org/abs/2504.19701
8. Mikhail Venchakov, Rook placements and coadjoint orbits for maximal unipotent subgroups of finite symplectic groups, arXiv:2602.14933 (2026), https://arxiv.org/abs/2602.14933
9. Mikhail Ignatev, Mikhail Venchakov, Orbits and characters associated with rook placements for Sylow $p$-subgroups of finite orthogonal groups, arXiv:2406.04436 (2024; v2 2025, accepted in Proceedings of the AMS), https://arxiv.org/abs/2406.04436
10. I. M. Isaacs, Counting characters of upper triangular groups, Journal of Algebra 315 (2007), 698-719, DOI 10.1016/j.jalgebra.2007.01.027, https://doi.org/10.1016/j.jalgebra.2007.01.027
11. Eric Marberg, Combinatorial methods of character enumeration for the unitriangular group, Journal of Algebra 345 (2011), 295-323, DOI 10.1016/j.jalgebra.2011.07.035, https://doi.org/10.1016/j.jalgebra.2011.07.035
12. Martin Marjoram, Irreducible characters of small degree of the unitriangular group, Irish Mathematical Society Bulletin 42 (1999), 21-31, https://www.maths.tcd.ie/pub/ims/bull42/marjoram.pdf
13. Igor Pak, Andrew Soffer, On Higman's $k(U_n(\mathbb F_q))$ conjecture, arXiv:1507.00411 (2015), https://arxiv.org/abs/1507.00411
14. Mikhail Ignatev, Leonid Titov, On the number of irreducible representations for finite unipotent Heisenberg-type groups, arXiv:2604.01358 (2026), https://arxiv.org/abs/2604.01358