Does this representation have a name? Let $G$ be a group acting on a set $X$. Let $F(X) = \{f: X \to \mathbb C\}$ be the set of complex valued functions on $X$. This is a complex vector space.
Then $G$ acts on $F(X)$ linearly via the following operation: 
$$ \pi (g) f(x) := f(g^{-1}x)$$
where $\pi : G \to (F(X)\to F(X))$ is the representation. Or at least that's what I gathered from the lecture. 
Now my question is: does this representation have a name? The lecturer called it a ''quantization of a group action'' but when I tried to google that I found nothing. 
 A: If a group $G$ acts on a set $X$, the induced action on $\mathbb{C}[X]$, the free complex vector space on $X$, is called the permutation representation associated to the $G$-set $X$. The induced action on $\mathbb{C}^X$ is the dual of the permutation representation. When $X$ is finite, these two representations are isomorphic, but when $X$ is infinite they don't even have the same dimension in general. (When $X$ is infinite the dual representation is probably not what you're looking for, and you want to cut it down in some way.)
Understanding permutation representations reduces quickly to the case that $X$ is transitive, hence that $X = G/H$ for some subgroup $H$. Then $\mathbb{C}[G/H]$ and $\mathbb{C}^{G/H}$ are two versions of the induced representation $\text{Ind}_H^G(1)$, where $1$ is the trivial representation of $H$. When $H$ has finite index in $G$, these two representations are isomorphic, but again, in general the dual representation is probably not what you're looking for, and you want to cut it down in some way.
