G abelian Lie group, $(\pi, V)$ a finite-dim unitary rep. $\exists $ mutually orthog. 1-dim invariant linear subspaces s.t $V= \bigoplus_i V_i$

Question

Let G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G.

I proved that $\pi \text{ is irreducible iff dim } V = 1\tag 1$

Using this I want to show that

There exist mutually orthogonal onedimensional invariant linear subspaces $V_1,..., V_n$ of $V $ such that $V = V_1 \oplus · · · \oplus V_n$.

My attempt:

I know that every finite-dimensional unitary representation can be decomposed in direct sum of irreducible subrepresentations $V_i\subseteq V: V=\bigoplus_{i} V_i$. How to show that they are mutually orthogonal? Isn't it possible to have a decomposition where they aren't? I am supposed to use (1) here, so the $V_i$'s are one dimensional but I am not sure how is that useful

Answer 1

Let $\{\pi(g):g\in G\}\subset GL(V)$. These are unitary operatots that also commute ($\pi(g_1)\pi(g_2)=\pi(g_1g_2)=\pi(g_2g_1)=\pi(g_2)\pi(g_1))$.

Moreover, as they are unitary, they are also unitarily diagnolizable. It is known that if two unitarily diagnolizable operators commute, then they are mutually unitarily diagnolizable, and by induction this is true for every finite set of unitary operatots that pairwise commute.

Now let $B=\{v_1,\ldots,v_n\}$ be a unitarily diagnolizing basis for all those operators. Then $V=\bigoplus span\{v_i\}$ is your desired decomposition.