For the natural representation $\pi(g): V \to V: v \mapsto gv$ of $G=SO(n,\Bbb R)$ on $V= \Bbb C^n$ I want to prove that the space End$_G(V)$ of intertwiners from $(V,\pi)$ to itself and the centralizer of $SO(n,\Bbb R)$ in $M(n,\Bbb R)$ coincide ($n\ge 3$ if that is relevant)
I have the following argument:
The centralizer $C$ of $SO(n, \Bbb R)$ is the set of elements $T\in M(n,\Bbb R)$ such that $Tg=gT$ $\forall g \in SO(n,\Bbb R)$.
End$_G(V)$ is exactly the set of endomorphisms $T$ such that $T\circ \pi (g)=\pi (g)T\forall g \in SO(n,\Bbb R)$ The representation is just the standard representation $\pi(g)=g$. This shows that the space of intertwiners and the centralizer coincide.
But I wonder, This equivalence of $C$ and End$_G(V)$ relies on each element in End$_G(V)$, being represented/isomorphic (? I don't know how should I best phrase it ) to the corresponding $n\times n$ matrix that represents the linear map.
Looking at the centralizer definition the matrix should be in $M(n,\Bbb R)$ but looking at the End$_G(V)$ definition since it's a map from $\Bbb C^n$ to $\Bbb C^n$,I think the matrix is in $M(n,\Bbb C)$, isn't it? What could guarantee that this matrix is actually in $M(n,\Bbb R)$ ? Unless I write the complex numbers as two-column vectors, but then it is not an action on $\Bbb C^n$ anymore but on $\Bbb R^{2n}$ and the matrix gets size $2n\times 2n$. Then I can not state the equivalence that I am after anymore. Could you help me clarify this?
Note : I need this argument to work to prove that $\pi$ is irreducible, for $(n\ge 3)$. Specifically, it is needed to show that End$_G(V) \subseteq \Bbb CId_V$ and apply Schur's lemma (see The natural representation of $SO(n)$ is irreducible for $n\ge 3$ )