Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known how to classify the $G$-equivariant linear maps between $V$ and $W$. Can anyone enlighten me as to how this works? I am most interested in the case where $\text{dim}(V) = {\text {dim}}(W)$ and the maps are isomorphisms.

Also, in the quantum setting, ie for $SL_q(N)$, does this classification pass over to a classification of comodule maps between the modules $V_q$ and $W_q$?

Lie Groups Beyond an Introduction, 2nd edition. – MTS Jul 12 '13 at 18:40