  We study the motion of a hole with internal degrees of freedom, introduced to
the zigzag magnetic ground state of Na$_2$IrO$_3$, by using the self-consistent
Born approximation. We find that the low, intermediate, and high-energy spectra
are primarily attributed to the singlet, triplet, and quintet hole
contributions, respectively. The spectral functions exhibit distinct features
such as the electron-like dispersion of low-energy states near the $\Gamma$
point, the maximum M-point intensity of mid-energy states, and the hole-like
dispersion of high-energy states. These features are robust and almost
insensitive to the exchange model and Hund's coupling, and are in qualitative
agreement with the angular-resolved photoemission spectra observed in
Na$_2$IrO$_3$. Our results reveal that the interference between internal
degrees of freedom in different sublattices plays an important role in inducing
the complex dispersions.
