Let $G$ be an algebraic group over the complex numbers, $H$ a normal algebraic subgroup. Let $L$ be a k-dimensional $G-$submodule of a $G-$ module $W$ with stabilizer exactly $H$, and let $\phi$ be the corresponding homomorphism $G \xrightarrow{\phi} GL(V)$.
I am trying to prove that the orbit of the character $\chi: H \xrightarrow{\wedge^k \phi|_H} GL(\wedge^k L)$ is finite, under the action of conjugation by $G$. My issue is that I showed that there is only one character in the orbit and onischik and vinberg would never do that, so I am trying to find the flaw in my argument.
The argument:
For any $l \in \wedge^k L \subset \wedge^k W$, the $g \cdot \lambda^k \phi|_H(h)v=det(\phi(g^{-1}hg)v=det(\phi(g^{-1}))det(\phi(h))det(\phi(g))v=det(\phi(h))v=\chi(h)v$ which shows that there is only one character in the orbit of $G$.
Motivation: I am trying to prove Chevalley's theorem by going through Onishchik's and vinberg's exercise-proof. I need to understand chevalley's theorem because I haven't understood it for a year and its about time.