Representations of Finite Lie Algebras and Geometry of Reductive Lie Algebras

In the joint paper [10], the authors established a connection between representations of a finite Lie algebra g0 over a finite field Fq and their algebraic extension g = g0 ⊗Fq k for k = F̄q , which particularly describes irreducible representations of g0 through irreducible representations of g. In this process, the so-called Fg-stable modules play a crucial role, which arise from the Frobenius-Lie morphism Fg on g, a defining mapping of g0 in g. In particular, fix g = Lie(G) for a connected reductive algebraic group G over k, defined over Fq , with corresponding Frobenius map F . Then the corresponding Frobenius-Lie morphism Fg naturally arises. One has a finite reductive Lie algebra g0 = gFg over Fq . In [27], it was proved that a closed conical subvariety X in the restricted nilpotent cone of g admits an Fq-rational structure if and only if X is the support variety of a stable g-module. As an application of this result, we show that the Zariski closures of nilpotent orbits in Lie algebras of classical groups admits Fq-rational structures under some restrictions on p, the characteristic of k.