Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 19F / a / 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 $\rho (g_1)\rho (g_2)=\rho (g_1{g}_2)$. 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 $W=V_n$. The representation is irreducible.