Jordan spectra of nilpotent derivations from scalar geometry

We determine the complete Jordan spectrum of nilpotent derivations of a nonabelian class-two Lie algebra I over a characteristic-zero field. We assume that its scalar bracket space has a decomposable basis over the ground field and that its reduced geometric decomposable locus admits no nontrivial additive-group action. A plane-preserver algebra and a hyperplane arrangement determine the horizontal rank set R; an integer criterion then decides every full Jordan type. Every admitted type occurs over the ground field, and the spectrum remains unchanged under field extension. For u = dim(I/Z(I)), d = dim I′, f = dim(Z(I)/I′), and r = max R, the greatest partition is (2^r, 1^(u-2r)) + (f) + (1^d), with componentwise addition. The full spectrum recovers R and is a dominance order ideal exactly when R has no gaps. For fixed free central dimension f, finite scalar loci with identical support-sum rank functions can have different spectra; support dimension six is minimal. A family has Jordan strata and commuting varieties whose component dimensions differ without bound. Geometric consequences include a smooth rational greatest-type stratum, a crepant resolution, and a classification of nilpotent linear spaces of largest dimension. Hall polynomials give uniform Jordan-stratum point counts at almost all places for number-field models.

Source repository: https://github.com/eulogica/jordan-spectra-nilpotent-derivations

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