In Jantzen's "Nilpotent Orbits in Representation Theory", in Section 8.18 (page 103), he derives (under certain conditions on the group $G$) a character formula for the coordinate ring $K[\mathcal{N}]$, where $\mathcal{N}$ is the variety of nilpotent elements in the Lie algebra $\mathfrak{g}=\textrm{Lie}(G)$. He then states that this character formula was first proved by Hesselink (Characters of the nullcone, Math. Ann., 252 (1980), 179-182, accessed here). He says that Hesselink's proof used the fact that $S(\mathfrak{g^*})\cong K[\mathcal{N}]\otimes S(\mathfrak{g}^*)^G$ and then adds parenthetically that Hesselink assumed char$(K)=0$ but that his proof generalizes.
The isomorphism $S(\mathfrak{g^*})\cong K[\mathcal{N}]\otimes S(\mathfrak{g}^*)^G$ was proved by Kostant in his "Lie group representations on polynomial rings" paper from 1963 and seemed to depend heavily on the base field being $\mathbb{C}$. So my question is, when Jantzen says that Hesselink's proof generalizes, does he mean that the isomorphism above holds in positive (or at least good) characteristic? If so, is there a source for this?