Spaltenstein's conjecture: every stabiliser sheet of a reductive Lie algebra contains a unique nilpotent orbit, in arbitrary characteristic

For a connected reductive algebraic group $G$ over an algebraically closed field of arbitrary characteristic, with Lie algebra $\mathfrak g$, the stabiliser sheets of $\mathfrak g$ are the irreducible components of the level sets $\mathfrak g^{(m)} = \{x \in \mathfrak g \mid \dim C_G(x) = m\}$ of the stabiliser-dimension map. Spaltenstein conjectured in 1982 that every such sheet contains exactly one nilpotent $G$-orbit. The conjecture is confirmed for groups with no exceptional-type simple components (all characteristics), for type $E_6$, for type $F_4$ in characteristic 2, in all good characteristics (Borho–Kraft), and — by Summerfield's 2025 extension of the decomposition-class machinery to arbitrary characteristic — for all Levi-type stabiliser sheets. The surviving open instances are $G_2$ in its bad characteristics (2 and 3), $F_4$ in characteristic 3, and $E_7$, $E_8$ in their bad characteristics — precisely the settings where centralizers of semisimple elements can be disconnected and the classical arguments degenerate. Resolving the conjecture would complete the classification of sheets and of the distribution of nilpotent orbits among them for all reductive Lie algebras, and would settle an infinitesimal-sheet analogue of the Kac–Weisfeiler property with consequences in modular representation theory.

Background

Let $k$ be an algebraically closed field of characteristic $p \geq 0$ and let $G$ be a connected reductive algebraic group over $k$ (an algebraic group with trivial unipotent radical, built from a central torus and simple factors), with Lie algebra $\mathfrak g = \mathrm{Lie}(G)$. A prime is good for $G$ if it divides no coefficient of the highest root of any simple component of the root system of $G$; all other primes are bad. Concretely, $p=2$ is bad for every irreducible root system except type $A$, $p=3$ is additionally bad for $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$, and $p=5$ is additionally bad for $E_8$. Most of the classical structure theory of $\mathfrak g$, including clean combinatorial parametrizations of nilpotent orbits, is available in good characteristic; bad characteristic is where it breaks down, and it is where this question lives.

Every $x \in \mathfrak g$ has a Jordan decomposition $x = x_s + x_n$ with $x_s$ semisimple, $x_n$ nilpotent, and $[x_s, x_n] = 0$. The nilpotent elements form the nilpotent cone $\mathcal N_{\mathfrak g}$, a closed $G$-stable cone, and $\mathcal N_{\mathfrak g}$ is a finite union of nilpotent $G$-orbits in every characteristic (by partition combinatorics for classical types, and by work of Holt–Spaltenstein and of Liebeck–Seitz for the exceptional types in bad characteristic).

The function $x \mapsto \dim C_G(x)$, the dimension of the centralizer of $x$ in $G$, is upper semicontinuous, so each level set

$$\mathfrak g^{(m)} = \{x \in \mathfrak g \mid \dim C_G(x) = m\}$$

is locally closed in $\mathfrak g$. Its irreducible components are the stabiliser sheets of $\mathfrak g$: the maximal irreducible varieties along which elements can move while the group-stabilizer dimension stays constant. Stratifying instead by $\dim \mathfrak c_{\mathfrak g}(x)$, the dimension of the Lie-algebra centralizer, gives the centraliser sheets; the two notions coincide whenever $\mathrm{Lie}(C_G(x)) = \mathfrak c_{\mathfrak g}(x)$ for all $x$ (for example under the standard hypotheses used by Premet–Stewart), but they differ in bad characteristic. This distinction matters: the conjecture below is stated for stabiliser sheets, and Summerfield observed that it is false for centraliser sheets — for $G = \mathrm{PGL}_2$ in characteristic 2 there is a non-empty centraliser level set containing no nilpotent orbit at all.

Over $\mathbb{C}$, Borho and Kraft showed that the sheet structure is governed by the decomposition classes (also called Jordan classes): the classes of the equivalence relation on $\mathfrak g$ that identifies $x = x_s + x_n$ with $y = y_s + y_n$ when the nilpotent parts are conjugate and the centralizers of the semisimple parts are conjugate compatibly. Each decomposition class has the form $G \cdot (z(\mathfrak l)^{\mathrm{reg}} + \mathcal O)$, where $L \subseteq G$ is a Levi subgroup (the reductive complement of the unipotent radical of a parabolic subgroup, e.g. a block-diagonal $\mathrm{GL}_{n_1} \times \cdots \times \mathrm{GL}_{n_r}$ in $\mathrm{GL}_n$), $\mathfrak l = \mathrm{Lie}(L)$, $z(\mathfrak l)$ is the centre of $\mathfrak l$, $z(\mathfrak l)^{\mathrm{reg}}$ its open subset of elements whose centralizer is exactly $L$, and $\mathcal O$ is a nilpotent $L$-orbit in $\mathfrak l$. Every sheet is a finite union of decomposition classes and contains a unique dense one. The key operation is Lusztig–Spaltenstein induction, which induces a nilpotent orbit from a Levi factor of a parabolic subgroup through the associated fibre bundle over the flag variety (the projective variety of Borel subgroups of $G$, i.e. its maximal connected solvable subgroups); a nilpotent orbit is rigid when it cannot be induced from a proper Levi subgroup. In characteristic zero (Borho) and in good characteristic (Premet–Stewart), the sheets of $\mathfrak g$ are in bijection with the $G$-conjugacy classes of pairs $(L, \mathcal O)$ with $\mathcal O$ a rigid nilpotent orbit of $\mathfrak l$.

Within this theory, a basic structural question is how the nilpotent orbits — the building blocks entering the Jordan decomposition — are distributed among the sheets. Spaltenstein proved that every sheet contains at least one nilpotent orbit, and in his 1982 study of Lie algebras in bad characteristic he conjectured that every sheet contains exactly one. The conjecture holds when $G$ has no exceptional-type simple components (all characteristics), for type $E_6$, for $F_4$ in characteristic 2, and in all good characteristics. Summerfield recently extended the decomposition-class machinery to arbitrary characteristic — including a dimension formula for decomposition classes and the parabolic independence of Lusztig–Spaltenstein induction of arbitrary orbits — and thereby confirmed the conjecture for all Levi-type stabiliser sheets. Here an element $x = x_s + x_n \in \mathfrak g$ is called Levi-type when $C_G^{\circ}(x_s)$ is a Levi subgroup of $G$; a decomposition class all of whose elements are Levi-type is strongly Levi-type, and a sheet is Levi-type when its unique dense decomposition class is a Levi-type decomposition class (Summerfield, arXiv:2511.01440, §7). What remains open are $G_2$ in its bad characteristics (2 and 3), $F_4$ in characteristic 3, and $E_7$, $E_8$ in their bad characteristics: exactly the cases where centralizers of semisimple elements can be disconnected, where $\mathrm{Lie}(C_G(x))$ can differ from $\mathfrak c_{\mathfrak g}(x)$, and where elements need not be Levi-type, so the classical arguments have no direct analogue.

The distribution of nilpotent orbits among sheets also has representation-theoretic content: Premet and Stewart note that the conjecture would imply an infinitesimal-sheet analogue of the Kac–Weisfeiler property (a dimension bound for the simple modules of the reduced enveloping algebra $U_\chi(\mathfrak g)$ — a quotient of the universal enveloping algebra depending on a linear form $\chi$ — attached to a nilpotent form $\chi$, i.e. one corresponding to a nilpotent element of $\mathfrak g$ under a $G$-equivariant identification $\mathfrak g \cong \mathfrak g^*$), which is one of the basic tools in the modular representation theory of $\mathfrak g$ and in the theory of modular finite $W$-algebras (the positive-characteristic analogues of the finite $W$-algebras attached to nilpotent elements).

Problem Statement

Let $G$ be a connected reductive algebraic group over an algebraically closed field of arbitrary characteristic, $\mathfrak g = \mathrm{Lie}(G)$, and for each $m \geq 0$ let

$$\mathfrak g^{(m)} = \{x \in \mathfrak g \mid \dim C_G(x) = m\}$$

where $C_G(x)$ is the centralizer of $x$ in $G$. The irreducible components of these locally closed level sets are the stabiliser sheets of $\mathfrak g$. Spaltenstein's conjecture asserts: every stabiliser sheet of $\mathfrak g$ contains exactly one nilpotent $G$-orbit.

Every stabiliser sheet is known to contain at least one nilpotent orbit, and the conjecture has been confirmed for: reductive groups with no exceptional-type simple components (all characteristics); type $E_6$; type $F_4$ in characteristic 2; all $G$ in good characteristic; and all Levi-type stabiliser sheets in arbitrary characteristic. The unresolved instances are the group $G_2$ in its bad characteristics (2 and 3), the group $F_4$ in characteristic 3, and the groups $E_7$, $E_8$ in their bad characteristics.

The problem is to settle these remaining instances in either direction: prove that every stabiliser sheet of $\mathfrak g$ contains exactly one nilpotent $G$-orbit in the remaining exceptional bad-characteristic cases (and hence for every connected reductive group over an algebraically closed field of arbitrary characteristic), or disprove the conjecture by exhibiting an explicit stabiliser sheet containing two or more distinct nilpotent $G$-orbits.

References

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2. G. Lusztig and N. Spaltenstein, 'Induced unipotent classes', Journal of the London Mathematical Society (2) 19 (1979), 41–52. DOI: 10.1112/jlms/s2-19.1.41. https://doi.org/10.1112/jlms/s2-19.1.41
3. W. Borho, 'Über Schichten halbeinfacher Lie-Algebren', Inventiones Mathematicae 65 (1981), 283–317. DOI: 10.1007/BF01389016. https://doi.org/10.1007/BF01389016
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9. A. Premet and D. I. Stewart, 'Rigid orbits and sheets in reductive Lie algebras over fields of prime characteristic', Journal of the Institute of Mathematics of Jussieu 17(3) (2018), 583–613. DOI: 10.1017/S1474748016000086. https://doi.org/10.1017/S1474748016000086
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12. J. Summerfield, 'Lie Algebra Decomposition Classes for Reductive Algebraic Groups in Arbitrary Characteristic', arXiv:2511.01440 (2025, v2 revised July 2026). DOI: 10.48550/arXiv.2511.01440. https://arxiv.org/abs/2511.01440
13. F. Ambrosio, G. Carnovale, F. Esposito, N. Saunders and L. Topley, 'Sheets, Jordan classes and induced orbits in the exotic and enhanced modules', arXiv:2602.02404 (2026). DOI: 10.48550/arXiv.2602.02404. https://arxiv.org/abs/2602.02404