Representation theory of SU(2)

Example

Let $G = \mathrm{SU}(2)$ be the group of complex $2 \times 2$ unitary matrices of determinant $1$. For each integer $n \ge 0$, let $V_n$ be the space of homogeneous polynomials of degree $n$ in two complex variables $z = (z_1, z_2)$, of dimension $n + 1$. Put $(\pi_n(g) P)(z) = P(g^{-1} z)$ for $g \in G$ and $P \in V_n$. This is a continuous homomorphism $G \to \mathrm{GL}(V_n)$, so it is a representation of $G$.

The classification: each $V_n$ is irreducible, the $V_n$ are pairwise non-isomorphic, and each irreducible finite-dimensional complex representation of $G$ is isomorphic to exactly one $V_n$.

Check of the character on $t_\theta = \operatorname{diag}(e^{i\theta}, e^{-i\theta})$, with $\theta \notin \pi\Z$. The monomial $z_1^a z_2^b$ with $a + b = n$ is an eigenvector with eigenvalue $e^{i(b - a)\theta}$, so $$\tr \pi_n(t_\theta) = \sum_{j=0}^{n} e^{i(n - 2j)\theta} = \frac{\sin\big((n+1)\theta\big)}{\sin \theta}.$$ For $n = 1$ this gives $2 \cos \theta$, the trace of $t_\theta$ itself. Since each element of $G$ is conjugate to some $t_\theta$, these characters determine the representations.