  We investigate harmonically-trapped, laser-pumped bosons with infinite-range
interactions induced by a dissipative high-finesse red-detuned optical cavity
with numerical and analytical methods. We obtain multiple cavity and atomic
observables as well as the full phase diagram of the system using the
multiconfigurational time-dependent Hartree method for indistinguishable
particles (MCTDH-X) approach. Besides the transition from an unorganized normal
phase to a superradiant phase where atoms self-organize, we focus on an
in-depth investigation of the self-organized superfluid to self-organized Mott
insulator phase transition in the superradiant phase as a function of the
cavity-atom coupling. The numerical results are substantiated by an analytical
study of an effective Bose-Hubbard model. We numerically analyze cavity
fluctuations and emergent strong correlations between atoms in the many-body
state across the Mott transition via the atomic density distributions and
Glauber correlation functions. Unexpectedly, the weak harmonic trap leads to
features like a lattice switching between the two symmetry-broken
$\mathbb{Z}_2$ configurations of the untrapped system and a reentrance of
superfluidity in the Mott insulating phase. Our analytical considerations
quantitatively explain the numerically observed correlation features.
