= Solution
For $z=(x,y)^T$, define $(\rho(g)f)(z)=f(g^{-1}z)$. This preserves the space of degree-$n$ <homogeneous polynomials> and satisfies $\rho(g_1)\rho(g_2)=\rho(g_1g_2)$. The diagonal torus of <SU(2)> acts on the monomials $x^{n-j}y^j$ with distinct weights $2j-n$.
If a nonzero subspace $W$ is invariant, averaging against each torus character projects $W$ onto the individual weight spaces. Therefore $W$ contains at least one monomial. Invariance also implies invariance under differentiated Lie-algebra operators and their complex linear combinations. The complexified algebra contains the operators $x\partial_y$ and $y\partial_x$, up to harmless signs. They carry each monomial to its adjacent monomial with a nonzero coefficient unless already at the appropriate endpoint. Repeated application produces every monomial, so $W=V_n$. \b[The representation is irreducible.]
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