Blokhuis showed that Paley graphs with square order have the Erdős-Ko-Rado (EKR) property in the sense that all maximum cliques are canonical. In the previous talk, Shamil discussed the extension of the EKR property of Paley graphs to certain Peisert graphs and generalized Peisert graphs, in particular, to certain pseudo-Paley graphs. In this talk, I will discuss the rigidity of maximum cliques in pseudo-Paley graphs from unions of semi-primitive cyclotomic classes. Our main result shows that in each maximum clique the contribution from each cyclotomic class must be equal. The key tools involved are Fourier analytic, namely Gauss sums and character sums. I will also discuss the subspace structure of maximum cliques and motivate a conjecture which generalizes the EKR property of Paley graphs, and can be viewed as an analogue of Chvátal's Conjecture for families of set systems. Joint work with Shamil Asgarli.