Linear perturbations of black hole backgrounds are relevant to a number of applications, from the ringdown phase of merger events to stability and AdS/CFT. The usual method to study the spectrum of these quasi-normal modes breaks down near the interesting extremal case. This hindrance was overcome recently by exploring the analogy between the differential equations that describe the perturbations and conformal blocks in conformal field theories. We use the analogy to explore scalar massive perturbations of the Kerr black hole near extremality, showing the existence of a exceptional point where the spectrum is degenerate and the existence of a geometrical phase as the eigenvalues are brought adiabatically around the point. We discuss these results using a thermodynamic analogy.