Suppose 𝐺𝕜 is a connected reductive algebraic 𝕜-group where 𝕜 is an algebraically closed field. If 𝑉𝕜 is a 𝐺𝕜-module then, using geometric invariant theory, Kempf has defined the nullcone 𝒩(𝑉𝕜) of 𝑉𝕜. For the Lie algebra 𝔤𝕜 = Lie(𝐺𝕜), viewed as a 𝐺𝕜-module via the adjoint action, we have 𝒩(𝔤𝕜) is precisely the set of nilpotent elements.
We may assume that our group 𝐺𝕜 = 𝐺 ×ℤ 𝕜 is obtained by base-change from a suitable ℤ-form 𝐺. Suppose 𝑉 is 𝔤 = Lie(G) or its dual 𝔤* = Hom(𝔤, ℤ) which are both modules for 𝐺, that are free of finite rank as ℤ-modules. Then 𝑉 ⨂ℤ 𝕜, as a module for 𝐺𝕜, is 𝔤𝕜 or 𝔤𝕜* respectively.
It is known that each 𝐺ℂ -orbit 𝒪 ⊆ 𝒩(𝑉ℂ) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ𝕜 ∈ 𝑉𝕜 for any algebraically closed 𝕜. In this talk, we will explain two ways in which we might want ξ to have “good reduction” and how one can find elements with these properties. We will also discuss the relationship to Lusztig’s special orbits.
This is on-going joint work with Adam Thomas (Warwick).