Q:

Some representation of $SU(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, could any one tell me how is that Linear Transformation look like explicitly?And would you please tell me how this is an representation of $SU(2)$? I need to know in detail. 
Let $U\in SU(2)$ is just a linear map on $\mathbb{C}^2$, Define a Linear Transformation
$\Pi_m:V_m\rightarrow V_m$ given by $[\Pi_m(U)f](z)=f(U^{-1}z)$ where $f(z)=a_0z_1^m+a_1z_1^{m-1}z_2+\dots +a_mz_2^m$, $z=(z_1,z_2)\in\mathbb{C}^2$
What I know about a representation of $SU(2)$ by definition   is, it must be a vector space $V$ together with a homomorphism $\phi:SU(2)\rightarrow GL(V)$

A:

The representation is the map $\Pi_m: SU(2) \to GL(V_m)$ where for $U \in SU(2)$, $\Pi_m(U)$ is the transformation that takes $f \in V_m$ to $f \circ U^{-1}$, i.e.
$ (\Pi_m(U)(f))(z) = f(U^{-1}(z))$. Note that if $f$ is a homogeneous polynomial of degree $m$ then $f \circ U^{-1}$ is also a homogeneous polynomial of degree $m$, e.g. 
if $f(z) = z_1^{m_1} z_2^{m_2}$ and $U^{-1} = \pmatrix{a & b\cr c & d\cr}$, 
$f \circ U^{-1} = (a z_1 + b z_2)^{m_1} (c z_1 + d z_2)^{m_2}$. 

