On Fixed Points of a Map Defined by Alain Connes

In this paper we study certain analogues of the map defined by Alain Connes, which follows an idea of Atiyah in trying to simplify the proof of Feit-Thompson theorem. It turns out that the the non-zero fixed points of the map can be characterized explicitly as an abelian group, and we can calculate some examples. It turns out that most fixed points are not one dimensional representations for these examples, and much work need to be done.