Flavour and CP symmetries in the inverse seesaw

We consider an inverse seesaw mechanism of neutrino mass generation in which the Standard Model is extended by $3+3$ (heavy) sterile states, and endowed with a flavour symmetry $G_f$, $G_f=\Delta (3 \, n^2)$ or $G_f=\Delta (6 \, n^2)$, and a CP symmetry. These symmetries are broken in a peculiar way, so that in the charged lepton sector a residual symmetry $G_\ell$ is preserved, while the neutral fermion sector remains invariant under the residual symmetry $G_\nu=Z_2 \times CP$. We study the concrete setup, where the Majorana mass term for three of the sterile states conserves $G_\nu$, while the remaining mass terms (i.e. couplings of left-handed leptons and heavy sterile states, as well as the Dirac-type couplings among the latter) do not break the flavour or CP symmetry. We perform a comprehensive analysis of lepton mixing for different classes of residual symmetries, giving examples for each of these, and study in detail the impact of the additional sterile states on the predictions for lepton mixing. We further confront our results with those obtained in the model-independent scenario, in which the light neutrino mass matrix leaves the residual symmetry $G_\nu$ intact. We consider the phenomenological impact of the inverse seesaw mechanism endowed with flavour and CP symmetries, in particular concerning effects of non-unitarity of the lepton mixing matrix (which strongly constrain the parameter space of the scenario), prospects for neutrinoless double beta decay and for charged lepton flavour violating processes.