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Hi Nick, I don’t know of any proof actually, elementary or not, but my background isn’t very strong. It also seems that rational classes of $G$ having character values equal to the character degrees of the centraliser (up to sign) occur quite frequently, not just in this cyclic case, but I don’t know of any precise statements along these lines.
This would seemingly follow from a proof of the observation that in your setup, the character values of irreps of $G$ on the conjugacy class of a nonzero element of $P$ are all $0$, $1$ or $-1$. I would be curious to see if this has an elementary proof too.
This is great, thanks. I would like to leave the question up for a little while to see if any affirmative results are known, if nothing, then I'll accept this answer.
Fantastic, this really clarifies whats going on. Do you know of a good reference that covers the results you're using here, like the exact sequences and such?
