I am confused by the wavevector in spectral function $A(\mathbf k,\omega)$. How to understand this $\mathbf k$ for a periodic structure? And how is it related to the $\mathbf k$ (in first Brillouin Zone) we use in the band structure? If there are multibands, then in band structure for the same k we have several energies. To roughly reproduce spectral function, do I expect several peaks for the same $\mathbf k$ in $A(\mathbf k,\omega)$ or I need to unfold the multibands, and the multi energies for same $\mathbf k$ in band structure appear at different $\mathbf k$ in $A(\mathbf k,\omega)$?
The spectral function has a peak function at the energy and momentum of long-lived excitations. As with all observables in a crystal the spectral function is fully defined by the first Brillouin Zone. In the case of the spectral function corresponding to the physical electron Green's function, you would see at fixed momentum a peak at the energy for every band. This is what is measured in ARPES experiments for example.