What is this $T^\#$ useful for / what is the meaning of $T^\#$? I'm reading Representation Theory of Finite Groups by Steinberg. In the chapter on characters, we have a proposition.

Proposition 4.2.2. Let $\varphi: G \to GL(V)$ and $\rho: G \to GL(W)$ be representations (of a finite group $G$). Suppose that $T:V \to W$ is a linear transformation. Then:


(a) $T^\#= \frac{1}{|G|} \displaystyle\sum_{g \in G} \rho_{g^{-1}}T\varphi_g \in $ Hom$_G(\varphi,\rho)$


(b) If $T \in$ Hom$_G(\varphi,\rho)$, then $T^\# = T$.


(c) The map $P:$ Hom$(V,W) \to$ Hom$_G(\varphi, \rho)$ defined by $P(T) = T^\#$ is a surjective linear transformation.


The proposition itself is meaty enough to deal with on its own, but I must ask, where does this $T^\#$ come from? I am having a hard time understanding why its derivation is relevant or how $T^\#$ came to be. Does anyone have any insight on what the actual meaning of this seemingly abstract $T^\#$ is or what it is useful for?
 A: This is my story on how $T^\sharp$ came to be.
You might be familiar with the averaging operator: if $\psi \colon G \to \operatorname{GL}(U)$ is any representation of the finite group $G$, then the operator
$$ \psi_1 := \frac{1}{|G|} \sum_{g \in G} \psi(g) $$
acts as a projector to the sum of all trivial subrepresentations of $U$. Indeed, if $u \in U$ is fixed by every element of $G$ then $\psi_1(u) = u$, and you can check that $\psi_1 \circ \psi_1 = \psi_1$.
You might also be familiar with the following construction: if $\varphi$ is a representation of $G$ on $V$, and $\rho$ is a representation of $G$ on $W$, then we can define another representation $\gamma$ of $G$ on $\operatorname{Hom}(V, W)$ by $\gamma(g, T) = \rho(g) \circ T \circ \varphi(g)$. A linear map $T \colon V \to W$ is an intertwining operator if and only if $T$ is fixed by every element of $G$, i.e. if $T$ is in the trivial subspace of $\operatorname{Hom}(V, W)$. This is very easy to check!
The $T^\sharp$ in your question is the projection operator $\gamma_1$ applied to the vector $T \in \operatorname{Hom}(V, W)$. So it is taking an arbitrary linear operator $T$ and projecting it to $T^\sharp$ in the subspace of intertwining operators $\operatorname{Hom}_G(V, W)$, i.e. the trivial subrepresentation of $\operatorname{Hom}(V, W)$.
