Spectral selectors on lens spaces and applications to the geometry of the group of contactomorphisms

Using Givental’s non-linear Maslov index we define a sequence of spectral selectors on the universal cover of the identity component of the contactomorphism group of any lens space. As applications, we prove for lens spaces with equal weights that the standard Reeb flow is a geodesic for the discriminant and oscillation norms, and we define for general lens spaces a stably unbounded conjugation invariant spectral pseudonorm.