Let $\mathcal{g}$ be a simple, finite-dimensional complex Lie algebra, and let $\mathcal{M}$ be a representative system of finite-dimensional simple $\mathcal{g}$-modules (up to isomorphism). Determine all finite-dimensional simple modules of $\mathcal{g}\otimes\mathbb{C}[x]$.
I know that $\mathcal{g}$ is a finite-dimensional semisimple lie algebra if and only if $\mathcal{g}$ has a faithful, finite-dimensional, simple representation. What I have already shown is that every ideal of $\mathcal{g}\otimes\mathbb{C}[x]$ is of the form $\mathcal{g}\otimes I$, where $I \subset \mathbb{C}[x]$ is an ideal. How can I proceed with this information? Any help is greatly aprreciated!
Edit: The bracket is, of course, defined by:$[a \otimes p(x), b \otimes q(x)]=[a,b]\otimes p(x)q(x)$ where $[a,b]$ represents the bracket of $\mathcal{g}$.
A representation of $\mathfrak g\otimes\mathbb C[x]$ is equivalent to a $\mathfrak g$-module $V$ together with a $\mathfrak g$-equivariant endomorphism $f\colon V\to V$. I claim the irreducible $\mathfrak g\otimes\mathbb C[x]$-modules are of the form: a irreducible $\mathfrak g$-module $V$ together with a scalar endomorphism $a\colon V\to V$.
Indeed, if $(W,f\colon W\to W)$ is an irreducible $\mathfrak g\otimes\mathbb C[x]$-module, then $W$ decomposes as $\bigoplus V_i^{n_i}$ as a $\mathfrak g$-module, where $V_i$ are pairwise non-isomorphic irreducible $\mathfrak g$-modules. Now by Schur's lemma $f$ restricts to endomorphisms $f_i\colon V_i^{n_i}\to V_i^{n_i}$, so $W$ must contain a unique $\mathfrak g$-representation $V_1$, i.e., $W\simeq V_1^{n_1}$. Now the endomorphism $f$ is represented by a $n_1\times n_1$-matrix, so by Jordan normal form we know in fact $n_1=1$.