Reading the figure
The four labels 1, 2, 4 and 8 form a cycle under multiplication by two modulo fifteen. The arrows encode the fixed geometric Frobenius convention.
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
1 2 4 8 × 2 mod 15
Original figure caption
For \(q=2\) modulo 15, the displayed geometric projector orbit is \((1,2,4,8)\).
For an irreducible tame Weil representation, a union of Frobenius orbits of nonzero inertia packets would give a stable subspace. Irreducibility therefore forces one orbit, of some length \(f\).
The stabilizer of a packet is \(W_{K_f}=\langle I_K,\Phi^f\rangle\), where \(K_f/K\) is the unramified extension of degree \(f\). Inertia acts on the packet by one scalar character. Its remaining stabilizer action is generated by \(\Phi^f\); an irreducible complex action of this cyclic quotient is one dimensional. Hence
Reading the diagram
Orbit transitivity supplies both the dimension and the induced model. It is a theorem about the tame case.