Base-case conjecture for centrally extended orbits in finite models

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Let ee be a nilpotent element in a classical Lie algebra g\mathfrak{g}, with partition λ\lambda, and let λ⊺\lambda^\intercal be its dual partition. Form λred\lambda_{\mathrm{red}} by removing 2l2l parts whenever a part of λ⊺\lambda^\intercal has multiplicity 2l+12l+1 or 2l+22l+2, and set λ′=λred⊺\lambda'=\lambda_{\mathrm{red}}^\intercal. Let e′e' be the corresponding nilpotent element in a Lie algebra of the same type, and let Ye,Ye′Y_e,Y_{e'} be the associated finite models; under the natural identifications Qe≅Qe′Q_e\cong Q_{e'} and Ae≅Ae′A_e\cong A_{e'}, compare their centrally extended orbits. Base-case conjecture. An QeQ_e-centrally extended orbit appears in YeY_e if and only if it appears in Ye′Y_{e'}. This predicts that the orbit structure for arbitrary partitions is determined by the reduced base cases; the supplied passage gives no resolution.

References

Primary source

Do Kien Hoang, “Geometry of the fixed points loci and discretization of Springer fibers in classical types”, arXiv:2405.10105 (2024).

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