Let $G$ be a Lie group and $X$ be a nonempty set. Let $f:G \rightarrow X$ be an injective function with range $\text{Im }f$ and left-inverse $f^{-1}$. Consider the $G$-action on $\text{Im}\; f$ defined by $a\cdot y := f(af^{-1}(y))$, for all $(a,y) \in G \times \text{Im }f$.
Indeed, this is a group-action because
- $e_G\cdot y = f(e_Gf^{-1}(y)) = f(f^{-1}(y)) = y$ forall $y \in \text{Im}\; f$.
- For $a,b \in G$ and $y \in \text{Im }f$, noting that $bf^{-1}(y) = f^{-1}(b\cdot y)$, we have
$$ \begin{split} (ab)\cdot y &:= f((ab)f^{-1}(y)) = f(a(bf^{-1}(y)))\\ &= f(af^{-1}(b\cdot y)) =: a\cdot(b \cdot y) \end{split} $$
Moreover, this action is a free group-action since all the stabilizers are trivial, i.e the set $\text{Stab}_G(y) := \{g \in G \mid a\cdot y = y\}$ is the singleton $\{e_G\}$ for all $y \in \text{Im }f$.
Question
- Have such group actions been studied before ?
- Do they have a canonical name ? Do they have any interesting properties ?
- Are the any interesting things which can been said or asked about such situation (i.e $G$ and $X$ and the above group action) ?
In representation theory,
I think that we can use such actions to define new representations.
When $V$ and $W$ are $k$-linear representations of a group $G$, you can define a representation on $\text{Hom}_{k}(V,W)$ using an action looking like yours and it is also the most naturel way to define an action: