Modular Reduction of Nilpotent Orbits

We consider a split connected reductive algebraic ℤ-group 𝐺 and 𝑉 a 𝐺-module which is either the Lie algebra 𝔤 or its dual 𝔤*. If 𝕜 is an algebraically closed field then, by base change, we get a group 𝐺𝕜 and a corresponding module 𝑉𝕜. Hesselink has defined a partition of the nullcone 𝒩(𝑉𝕜) of 𝑉𝕜 into strata 𝒩(𝑉𝕜 | 𝒪) which can be indexed, thanks to Clarke–Premet, by 𝐺(ℂ)-orbits 𝒪⊆ 𝒩(𝔤ℂ), such that 𝒩(𝔤ℂ | 𝒪) = 𝒪 . Each stratum is a union of 𝐺(𝕜)-orbits.In this talk I will describe joint work with Adam Thomas (Warwick) which produces for each orbit 𝒪⊆ 𝒩(𝔤ℂ), via a case-by-case analysis, integral representatives 𝑒 ∈ 𝑉∩𝒩(𝑉ℂ | 𝒪) whose reduction 𝑒𝕜 ∈ 𝒩(𝑉𝕜 | 𝒪) is well-behaved for every algebraically closed field 𝕜. There are three possibilities for what well-behaved can mean and we treat all three.