We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
We denote
$$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$
Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.