Write $H^k(\mathbb{C}^2)$ for the space of harmonic polynomials on $\mathbb{C^2}$ of degree $k\in \mathbb{Z}_{\geq 0}$. It can be easily shown that $$H^k(\mathbb{C}^2) = \mathbb{C}(z_1 + i z_2)^k + \mathbb{C}(z_1 - iz_2)^k$$ This space is an $O(2)$-module with the standard action given by $p \mapsto p(B^{-1}\cdot)$, for $B\in O(2)$.
We also consider the $O(2)$-module $\mathbb{C}^2_k$ defined by $O(2) \rightarrow GL_\mathbb{C}(\mathbb{C}^2), B \mapsto B^k$.
I know that this two $O(2)$-modules are isomorphic, that is $H^k(\mathbb{C}^2) \simeq \mathbb{C}^2_k$, but I don't know how to prove it.
The only case I could prove is the trivial one, namely the case $k = 1$. For this case, the map below is an $O(2)$-equivariant isomorphism. \begin{equation*} \begin{gathered} H^1(\mathbb{C}^2) \rightarrow \mathbb{C}^2_1\\ a(z_1 + iz_2) + b(z_1 - iz_2) \mapsto a\begin{pmatrix}1\\i\end{pmatrix}+ b\begin{pmatrix}1\\-i\end{pmatrix} \end{gathered} \end{equation*}
When I try to do something similar for $k\geq 2$, the $O(2)$-equivariance fails.
Maybe I'm missing some facts of representation theory. Do you have any idea?
Edit: I managed to prove that the same map in arbitrary degree, that is \begin{equation*} \begin{gathered} H^k(\mathbb{C}^2) \rightarrow \mathbb{C}^2_k\\ a(z_1 + iz_2)^k + b(z_1 - iz_2)^k \mapsto a\begin{pmatrix}1\\i\end{pmatrix}+ b\begin{pmatrix}1\\-i\end{pmatrix} \end{gathered} \end{equation*} Is an $SO(2)$-equivariant isomorphism, but I still have problems to prove the equivariance for reflections.