This a lemma used in (Scott 2006) when discussing complex projective t-designs.
Let $\pi(x)\equiv|x\rangle\!\langle x|$ be the projection onto some pure state (represented as an element of the complex projective space), denote with $\mu$ the uniform measure in the complex projective space, and let $\Pi_{\rm sym}^{(t)}$ be the projector onto the totally symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$. Then $$\int_{{\Bbb CP}^{d-1}}d\mu(x)\pi(x)^{\otimes t}=\binom{d+t-1}{t}^{-1} \Pi_{\rm sym}^{(t)}.\tag1$$ The author mentions that can be proved as follows:
Use Schur's lemma. The LHS of the equation is invariant under all unitaries $U^{\otimes t}$ which act irreducibly on the totally symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$.
While explanation makes sense, I'm looking to see some of the underlying details be written down more explicitly. In particular, we seem to be considering the group action of $\mathbf U(d)$ on $\mathbb{CP}^{d-1}$. I guess this is the action $(U,[v])\mapsto [Uv]$ for $U\in\mathbf U(d)$ and $v\in \mathbb{C}^d$, $[v]\in\mathbb{CP}^{d-1}$. This action is the one used to define the measure $\mu$, which is characterised by its being invariant under this action.
But then, we consider the action of $\mathbf U(d)$ on the totally symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$, and using Schur's lemma with this action. I'm assuming the version of Schur's lemma we're using is the fact that, given a complex finite-dimensional irreducible representation $\rho:G\to\operatorname{Lin}(\mathbb{C}^n)$, if a linear operator $f:\mathbb{C}^n\to \mathbb{C}^n$ is such that $[f,\rho(g)]=0$ for all $g\in G$, then $f=\lambda I_n$. I know that one way to use this lemma is to build explicitly elements that commute with the irrep by taking averages, and using this observation we get results of the form: $$\sum_{g\in G} \rho(g)A \rho(g)^\dagger = \frac{|G| \operatorname{Tr}(A)}{n} I_n,$$ for any linear operator $A:\mathbb{C}^n\to\mathbb{C}^n$.
This is clearly very close to (1), especially observing the fact that $\binom{d+t-1}{t}$ is the dimension of the totally symmetric subspace we're using, and $\Pi_{\rm sym}^{(t)}$ is essentially the identity, when we're restricting the domain on said space. Still, the LHS of (1) is averaging over the elements of another set over which $\mathbf U(d)$ acts, rather then over the group elements themselves, which is what is used in Schur's lemma, so what gives?