Let $\phi_1, \phi_2$ be representations of a compact Lie group $G \to V$ that are equivalent. Suppose furthermore that they are irreducible. It follows that there is an isomorphism, $T$, so that: $T \circ \phi_1 = \phi_2 \circ T$.
Schur's lemma states that $T = cI$ for some $c$. Dividing through by $c$ gives $\phi_1 = \phi_2$. This doesn't seem correct. Where am I going wrong?
In the form that you are applying it, Schur's lemma only applies to intertwining maps from a representation to itself. Here the two representations have the same underlying vector space $V$ but you need them to actually be the same representation, which just means that $\varphi_1=\varphi_2$.