As the title states, what does this notation mean? $(Hom_{\mathbb{C}}(V,W))^G$
$V, W$ are $G$ - spaces, where $G$ is a group.
It appeared in the following context;
$(Hom_{\mathbb{C}}(V,W))^G$ is equal to the space of all $G$-space homomorphisms $ V \rightarrow W$
$G$ acts on $\textrm{Hom}_{\Bbb C}(V,W)$ by $$(f\cdot g)(v)=f(v\cdot g^{-1})\cdot g.$$ $U^G$ is the set of $G$-invariants in $U$: ${U^G}=\{u\in U:u\cdot g=u\ \forall g \in G\}$. Thus $\textrm{Hom}_{\Bbb C}(V,W)^G$ is the set of all $f\in \textrm{Hom}_{\Bbb C}(V,W)$ such that $f\cdot g=f$ for all $g\in G$. This is equivalent to $f(v\cdot g)=f(v)\cdot g$ for all $v\in V$ and $g\in G$, that is to $f$ respecting the actions of $G$.