The Dirac spectrum for Sn is well known along with its multiplicities. In Appendix D of https://arxiv.org/pdf/1510.05663.pdf author computes the Dirac spectrum of RP4 from that of S4. The argument author uses is that RP4 has two pin+ structures and applying parity condition on Dirac spectrum of S4 gives the spectrum for RP4. My question is
1. Why parity condition and how does it relate pin and spin structures?
2. Suppose, I want to obtain the Dirac spectrum for RPn. I know n= 1 is easy as it has two inequivalent spin structures same as S1. But how do I obtain it for n=2,3,5 etc. I also know RP3 is orientable (for odd n) and has 2 pin+ and 2 pin− structures. But how is its spectrum different from that of S3?