Let $G$ be a reductive group over an algebraically closed field $k$ of characteristic $p>0$. If $V$ is a rational $G$-module then we can define the Hilbert nullcone $\mathcal{N}(V)$ to be the zero locus of all positive degree homogeneous invariant polynomials. By Haboush's theorem you can regard this as the trivial fibre of the quotient map $V \to V// G$. Let $S$ be the set of all points $v\in V$ such that the $G$-orbit of $v$ is $k^\times$-stable, where the scalar action is coming from the vector space structure.
Since the $k^\times$-action permutes the non-trivial fibres of $V \to V// G$ one can show that $S \subseteq \mathcal{N}(V)$. If $G$ acts with finitely many orbits on $\mathcal{N}(V)$ then you can show that $S = \mathcal{N}(V)$, by repeating the argument of Lemma 2.10 from Jantzen's book ``Nilpotent Orbits in Representation Theory''. In general however this argument doesn't work.
I have two questions:
1) does $S = \mathcal N(V)$ in general?
2) if not then is $S$ always a closed subvariety of $V$?
Thanks in advance for answers!
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Edit: I'm especially interested in the representation $\operatorname{Hom}(V \otimes V, V) \cong (V \otimes V)^* \otimes V$.