In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block.
Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = \text{Symm}^j(\mathbb{C}^2)$ be the $j$th symmetric power of the standard representation of $\mathbb{H}^{\times}\hookrightarrow \text{GL}_2(\mathbb{C})$ (hence a representation of $D^{\times}$). Note that dim$(V_j) = j+1$.
Now I know that $\mathbb{H}^{\times}$ acts on the space $\mathcal{P}_j$ of real harmonic homogeneous polynomials of degree j in $4$ variables (by viewing these polynomials as in one quaternion variable and changing variables by left multiplication). Note that dim$(\mathcal{P}_j) = (j+1)^2$.
My question is how can I view $V_j$ as an irreducible subrepresentation of $\mathcal{P}_j$?
(The reason for wanting to do this is due to the fact that when considering the transfer from modular forms on $D^{\times}$ to elliptic modular forms I have to consider theta series weighted by harmonic polynomials. In order to go from "weight $j$" on the left to "weight $j+2$" on the right I require real harmonic polynomials in $4$ variables).
