Representations of $\mathfrak{u}(1)$ on $\mathbb{C}$ and $\bar{\mathbb{C}}$ with winding number $k\neq0$ are not isomorphic.

Question

In the text I'm reading, $U(1)$ has a representation on $\mathbb{C}$ with winding number $k$ given by: $$z\mapsto z^k$$ I'm supposed to show that the representations of $\mathfrak{u}(1)$ with winding number $k$ on $V_k\cong\mathbb{C}$ and $\bar{V}_k$ are not isomorphic. I started by defining my representation of $U(1)$ on $V_k$ as: $$\Phi_{V^k}:e^{i\theta}\rightarrow e^{ik\theta}$$ I then said that the induced represenation of $\mathfrak{u}(1)$ on $V_k$ would be : $$\Phi_{V^k*}:X\rightarrow ikX$$ where $X\in \mathbb{R}$ since the Lie algebra of $U(1)$ is just $i\mathbb{R}$. So now I just need to show that a $g$ equivariant isomorphism from $V_k$ to $\bar{V}_k$ does not exist, but if I define the isomorphism $F:V_k\rightarrow \bar{V}_k$ such that: $$F(a)=\bar{a}$$ how is this not $g$ equivariant? Wouldn't $(ikX)\cdot (a)$ just get mapped to $-ikX \bar{a}$ which would be $\Phi_{\bar{V}^k*}(X)F(a)$? I'm failing to see what I'm missing.