Let \(t\) generate a finite tame quotient of order \(M\), with \((q,M)=1\). For \(j\in\mathbb Z/M\mathbb Z\) define
The finite Fourier identities give \(P_jP_k=\delta_{jk}P_j\) and \(\sum_jP_j=I\). Thus \(P_jV\) is the \(\zeta_M^j\) character space for \(r(t)\).
The guide uses geometric Frobenius with
Equivalently \(\Phi r(t)\Phi^{-1}=r(t)^{q^{-1}}\), where the inverse is taken modulo \(M\). Substituting in the Fourier sum gives
Scope
If the forward relation \(\Phi r(t)\Phi^{-1}=r(t)^q\) is used instead, the projector label is \(q^{-1}j\). The two directions have the same orbit sets but different conjugation identities. The release corrects the historical mixed-convention sentence.
Reading the diagram
A packet is an actual eigenspace selected by an idempotent. Frobenius carries it to the next character packet.