Jibladze and Kac (2021) proved a normal form theorem for nilpotent elements of complex semisimple Lie algebras, generalizing the Jordan normal form in slN\mathfrak{sl}_N, and used it to attach to every nilpotent element ff a conjugacy class wfw_f in the Weyl group WW. The paper explicitly defers the comparison of f↦wff\mapsto w_f with the Kazhdan–Lusztig map to a subsequent publication and asserts the even-depth analogue of its minimal reducing subalgebra theorem only conditionally and without proof. First, the map f↦wff\mapsto w_f coincides with the classical Kazhdan–Lusztig map from nilpotent orbits to conjugacy classes in WW on nilpotents of regular semisimple type but differs in general. Second, the analogue of the minimal reducing subalgebra theorem for nilpotent elements of even depth dd is asserted in Remark 2.1 only conditionally, on the existence of a non-zero Cartan subspace C⊆gd−1C\subseteq\mathfrak{g}_{d-1} for the action of the triple centralizer Z(s)Z(s), and that conditional analogue has not been established.

Background

Let g\mathfrak{g} be a complex semisimple Lie algebra. A nilpotent element f∈gf\in\mathfrak{g} can be embedded in an sl2\mathfrak{sl}_2-triple s=⟨e,h,f⟩\mathfrak{s}=\langle e,h,f\rangle, and the adjoint action of hh gives the Dynkin Z\mathbb{Z}-grading g=⨁j∈Zgj\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_j; the largest jj with gj≠0\mathfrak{g}_j\neq 0 is the depth dd of ff. An element of the form f+Ef+E with a non-zero E∈gdE\in\mathfrak{g}_d is called a cyclic element associated to ff; ff is of nilpotent type if all cyclic elements associated to ff are nilpotent, of semisimple type if some cyclic element associated to ff is semisimple, and of mixed type otherwise. The reduced depth d~\tilde d is the depth of ff in the even part gev=⨁j∈Zg2j\mathfrak{g}^{\mathrm{ev}}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{2j} of the grading; equivalently d~=d−1\tilde d=d-1 when ff is of nilpotent type and d~=d\tilde d=d when ff is of semisimple or mixed type. By work of Elashvili–Kac–Vinberg (2013) on cyclic elements, one is led to the notion of a reducing subalgebra for ff: a semisimple subalgebra q⊆g\mathfrak{q}\subseteq\mathfrak{g}, normalized by s\mathfrak{s}, such that Z(s)(q∩gd~)Z(s)(\mathfrak{q}\cap\mathfrak{g}_{\tilde d}) contains a non-empty Zariski open subset, where Z(s)Z(s) is the centralizer of the triple s\mathfrak{s} in the adjoint group; q\mathfrak{q} is the semisimple subalgebra to which the study of the cyclic elements associated to ff reduces. Elashvili–Kac–Vinberg conjectured that every non-zero nilpotent ff has a unique minimal reducing subalgebra up to Z(s)Z(s)-conjugacy, and Jibladze–Kac (2021) proved it (their Theorem 2.1): for a Cartan subspace C⊆gd~C\subseteq\mathfrak{g}_{\tilde d}, the subalgebra q(f,C)\mathfrak{q}(f,C) generated by ff and CC is a minimal reducing subalgebra when ff is of semisimple type, while for ff of nilpotent or mixed type it is the direct sum of a minimal reducing subalgebra with a one-dimensional center spanned by a nilpotent element. Here a Cartan subspace is the key object from the theory of polar representations introduced by Dadok and Kac (1985): for a reductive linear group GG acting on VV, a subspace C⊂VC\subset V all of whose points have closed GG-orbits, which meets every closed orbit finitely, with all such CC conjugate. A polar structure on Z(s)∣gd~Z(s)|\mathfrak{g}_{\tilde d} was verified case-wise by Elashvili–Jibladze–Kac (2022), and De Sole–Jibladze–Kac–Valeri (2021) proved, again case-wise, that Z(s)∣gd−1Z(s)|\mathfrak{g}_{d-1} is polar as well. Iterating Theorem 2.1 yields the normal form (Theorem 2.2 of Jibladze–Kac): every non-zero nilpotent ff is a sum f=∑jf[j]f=\sum_j f^{[j]} of commuting irreducible nilpotent elements (those whose only reducing subalgebra is the ambient algebra) in semisimple subalgebras of strictly decreasing depths. For an irreducible ff of regular semisimple type (meaning some cyclic element f+Ef+E associated to ff is regular semisimple), an inner automorphism σf\sigma_f of g\mathfrak{g}, of order mm or m/2m/2 and built from the Dynkin labels of ff and ε=e2πi/m\varepsilon=e^{2\pi i/m}, acts on a Cartan subalgebra centralizing f+Ef+E and thereby defines an element wfw_f of the Weyl group WW of g\mathfrak{g} (Table 8 of the paper lists them); the normal form extends this multiplicatively to all nilpotent elements. On the other hand, Kazhdan and Lusztig (1988) constructed, via fixed-point varieties on affine flag manifolds, a map from nilpotent orbits of g\mathfrak{g} to conjugacy classes in WW — the Kazhdan–Lusztig map — subsequently computed in tables by Spaltenstein (1990). Jibladze and Kac observed that their map f↦wff\mapsto w_f coincides with the Kazhdan–Lusztig map on nilpotents of regular semisimple type, but differs from it in general, and deferred the comparison to a subsequent publication. Separately, their Remark 2.1 states that for nilpotent elements of even depth dd an analogue of Theorem 2.1 should hold with the Cartan subspace taken in gd−1\mathfrak{g}_{d-1} rather than gd~\mathfrak{g}_{\tilde d} — but only conditionally: provided a non-zero Cartan subspace C⊆gd−1C\subseteq\mathfrak{g}_{d-1} exists, the subalgebra q(f,C)\mathfrak{q}(f,C) is either semisimple or the direct sum of a semisimple algebra with a one-dimensional center spanned by a nilpotent element, and correspondingly f+Ef+E is semisimple for generic E∈gd−1E\in\mathfrak{g}_{d-1}, or never semisimple. The assertion is given without proof.

Problem Statement

Resolve the two directions that Jibladze and Kac (2021) left unresolved in their normal form theory of nilpotent elements of complex semisimple Lie algebras, which together complete that program. (1) Determine the precise relationship between the normal-form map f↦wff\mapsto w_f (defined for irreducible nilpotents via the inner automorphism σf\sigma_f acting on a Cartan subalgebra, and extended multiplicatively through the normal form decomposition f=∑jf[j]f=\sum_j f^{[j]}) and the Kazhdan–Lusztig map from nilpotent orbits to conjugacy classes in the Weyl group WW. In particular, determine the criterion, for each nilpotent orbit of each simple Lie algebra, for the two maps to agree — equivalently, the orbits on which they agree and those on which they fail — and, for the orbits on which they fail, describe the discrepancy and explain, in representation-theoretic or geometric terms, why the criterion fails there, relating wfw_f to the Kazhdan–Lusztig data used by Spaltenstein's tables. (2) Establish, for nilpotent elements of even depth dd, the analogue of the minimal reducing subalgebra theorem that Jibladze–Kac assert only conditionally in Remark 2.1: prove that, when a non-zero Cartan subspace C⊆gd−1C\subseteq\mathfrak{g}_{d-1} for the polar action of Z(s)Z(s) exists, the subalgebra q(f,C)\mathfrak{q}(f,C) generated by ff and CC is either semisimple or the direct sum of a semisimple algebra and a one-dimensional center spanned by a nilpotent element, with f+Ef+E semisimple for generic E∈gd−1E\in\mathfrak{g}_{d-1} in the first case and never semisimple in the second.

References

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