Decompose the representation $V$ of $SO_2$ into irreducible representations Let $V=\mathbb{C^2}$ be the standard representation of $SO_2$ > Decompose $V$ into irreducible representations The standard unit vectors of $\mathbb{C^2}$ are $e_1$ and $e_2$ I am not sure how to use these to decompose $V$, does it have something to do with finding eigenvectors?

A way to do this is to use the Lie algebra: if $$ J = \pmatrix{ 0& -1\\\1 & 0 } = {d\over d \theta }{\Large|}_{\theta = 0 }\pmatrix{ \cos \theta & -\sin \theta\\\ \sin \theta & \cos \theta},$$ then $$J^2 + 1 = 0.$$ Diagonlize $J$: then the rotation $\exp J\theta $ will act on (stabilize) the eigen-spaces of $J$.

* btw: $ \exp J\theta$ acts by multiplication by $e^{i\theta}= \cos \theta + i \sin \theta$ on one eigen-space, and by multiplication by $e^{-i\theta}= \cos \theta - i \sin \theta$ on the other.