  Orbital degree of freedom plays an important role in understanding exotic
phenomena of strongly correlated materials. We study strongly correlated
ultracold bosonic gases coupled to a high-finesse cavity, pumped by a
blue-detuned laser in the transverse direction. Based on an extended
Bose-Hubbard model with parameters adapted to recent experiments, we find that
by tuning the reflection of pump laser, atoms can be selectively transferred to
the odd-parity $p$-orbital, or to even-parity $d$-orbital band of a
two-dimensional square lattice, accompanied with cavity-photon excitations. By
interacting with cavity field, atoms self-organize to form stable
higher-orbital superfluid and Mott-insulating phases with orbital-density
waves, as a result of cavity induced orbital-flip processes. Our study opens
the route to manipulate orbital degrees of freedom in strongly correlated
quantum gases via coupling to optical cavities.
