On counting orbits in root systems

The computation of the characters of supercuspidal representations of a p-adic group involves some 4th roots of unity whose values are defined in terms of orbits of the Galois group of a p-field on a root system. The part of the definition that is of interest in the verification of stability of character sums involves just the parity of the number of Galois orbits. In this paper, we re-cast the definition (nearly) in terms only of the abstract action of a pair of automorphisms on a root system, and compute it by a series of reductions in all cases.