  The energy band structure of excitons is studied in periodic potentials
produced by the short-range interaction between the exciton and electrons of
Wigner or Moir\'{e} lattices. Treating the exciton as a point-like dipole that
interacts with the periodic potential, we can solve a simple one-body problem
that provides valuable information on excitons in many-body problem settings.
By employing group theory, we identify the excitonic energy bands that can
couple to light and then quantify their energy shifts in response to a change
in the period of the potential. This approach allows us to emulate the response
of optically active exciton and trion states to a change in electron density.
We gain important insights on the relation between the electron order in a
Wigner crystal and the energy blueshift of the bright exciton. We discuss the
consequences of this relation in the context of optical absorption experiments
in monolayer semiconductors.
