Let $G$ be a connected reductive group over an algebraically closed field $k$ with Weyl group $W$. Let $$ \mathcal{S} = R\pi_*\mathbb{Q}_\ell[\dim \mathcal{N}] $$ be the Springer sheaf, where $\mathcal{N}\subset \mathrm{Lie} (G)$ is the nilpotent cone and $\pi:\widetilde{\mathcal{N}} \rightarrow\mathcal{N}$ is the Springer resolution.
In Zhiwei Yun's lecture notes, the proof of the Springer correspondence contains the following passage:
We decompose the perverse sheaf $\mathcal{S}$ into isotypical components under the $W$-action $$ \mathcal{S} = \bigoplus_{\chi\in \mathrm{Irr}(W)} V_\chi\otimes S_\chi $$ where $V_\chi$ is the space where $W$ acts by the irreducible representation $\chi$, and $\mathcal{S}_{\chi} = \hom_W(V_\chi, \mathcal{S})$ is a perverse sheaf on $\mathcal{N}$. Since $\mathrm{End}(\mathcal{S})\cong\mathbb{Q}_\ell[W]$, we conclude that each $\mathcal{S}_\chi$ is nonzero and that $$ \hom(\mathcal{S}_\chi,\mathcal{S}_{\chi'}) = \begin{cases} \mathbb{Q}_\ell & \chi = \chi' \\ 0 & \mathrm{otherwise}. \end{cases} $$
I am not sure how to obtain the bolded conclusion.
I believe it should be possible to compute $\mathrm{End}(\mathcal{S})$ in terms of the multiplicity sheaves $\mathcal{S}_\chi$. At this point in the proof, it is not yet known that the $\mathcal{S}_\chi$ are semisimple, so I do not know how to proceed in this direction.
In fact, I think the corresponding statement for complex representations is false. I.e., if $R$ is a finite dimensional $\mathbb{C}$-representation of $W$ with $\mathrm{End}(R) = \mathbb{C}[W]$, then the multiplicity of each $W$-irrep in $R$ is precisely 1. But $\mathrm{End}(\mathbb{C}[W]) \cong \mathbb{C}[W]$ and $\mathbb{C}[W]$ contains irreps with larger multiplicities.