TITLE: Is there a name for this representation-theoretic function?
QUESTION [6 upvotes]: Let $(\rho, V)$ and $(\tau, W)$ be two representations of a finite group $G$, and $h : V\to W$ a linear map, not necessarily an intertwining operator. Is there a name for the quantity
$$
h_G = \frac1{|G|}\sum_{t\in G}\tau(t^{-1})\circ h\circ\rho(t)\ ?
$$
In my notes I've dubbed this the "intertwinization" of $h$, as $h_G$ is easily checked to be an intertwining operator itself:
$$
\begin{aligned}
\tau(s^{-1})\circ h_G\circ\rho(s) & = \frac1{|G|}\sum_{t\in G}\tau((ts)^{-1})\circ h\circ\rho(ts) \\
& = \frac1{|G|}\sum_{t\in G}\tau(t^{-1})\circ h\circ\rho(t) = h_G
\end{aligned}
$$
I have a hunch this quantity shows up frequently. Is there any standard name for it? The weighted average of conjugates of $h$ I suppose?

REPLY [5 votes]: In any representation $\rho : G \to V$ we have an averaging operator $\frac{1}{|G|} \sum_{g \in G} \rho(g)$ which computes the projection to $V^G$. Here the representation is the internal hom $[V, W]$, and $[V, W]^G$ is the space of intertwining maps $V \to W$; this is part of a classic proof of the orthogonality relations for characters. (We need to be working over a field of characteristic not dividing $|G|$.)
The averaging operator is sometimes called the Reynolds operator, although that term also refers to more sophisticated variants involving e.g. averaging over a compact group. I think "averaging operator" is fine.