Corollary 3

Let V and $V'$ be finite dimensional vector spaces, and let $r: S\rightarrow$ End(V) and $r': S\rightarrow$ End(V$'$) be maps satisfying condition (AC). Let $f: V\rightarrow V'$ be a surjective linear map such that $f(r(s)v) =r'(s)f(v)$ for $s\in S$ and $v\in V$. Then $f(V^{\lambda}{(S)) = V'}^{\lambda}(S)$ for all $\lambda \in P$.

In view of Prop. 1, we are reduced to the case in which $k$ is algebraically closed. We have $V =\bigoplus_{\lambda\in P}V^{\lambda}$(S), $V'=\bigoplus_{\lambda\in P}{V'}^{\lambda}(S)$ by Th. 1, and $V'=f(V) =$ $\sum_{\lambda\in P}f(V^{\lambda}(S))$. Finally, $f(V^{\lambda}(S))\subset {V'}^{\lambda}(S)$ by Prop. 2 (i), hence the corollary.