Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group with the same root datum as $G_{\mathbb{C}}$. Let $\mathfrak{g}_k$ denote the corresponding Lie algebra.
Question: Suppose the characteristic of $k$ is large enough. Is it true that nilpotent orbits in $\mathfrak{g}_{\mathbb{C}}$ are in (canonical?) bijection with nilpotent orbits in $\mathfrak{g}_k$?
My impression is that various classification theorems (such as Bala-Carter or Dynkin) which are classical for $\mathbb{C}$ work equally well if $k$ has large characteristic. This indicates that the answer to above question is YES. Am I right? If so, can one give a proof that avoids the classification theorems?
