I am studying Lie groups, precisely their representations, and I am having troubles with understanding one part of the proof on Theorem concerning representations of $SU(2)$.
Namely, there has been said that for the sake of proof to show that a certain representation $V$ is irreducible, it is enough to show that each $SU(2)$-equivariant endomorphism $A$ of a given representation $V$ is a multiple of the identity.
Now, the thing is that this is sufficient to show, since otherwise you can take the projection onto a nontrivial submodule and it should work since this projection would be a nonscalar equivariant endo.
Can someone help me understanding why this projection should be equivariant, i.e. why does the following must hold:
$\forall g \in SU(2)$ and $\forall x \in V$ $gp(x)=p(gx)$, where $p$ is given projection?
It can be shown as follows: suppose satisfied the hypothesis and let $V$ a non zero invariant module. You can find a scalar product $b$ invariant by $SU(2)$ on the image $E$ of the representation since $SU(2)$ is compact. The orthogonal $W$ of $V$ is invariant and the morphism whose restriction to $V$ is $cId_V$ and to $W$ is $dId_W$ is invariant and is an isomorphism where $c,d$ are non zero real, this implies that $W=0$ since this morphism is a multiple of the identity.