We study the Fermi edge singularity in the absorption and emission spe
ctra of quantum Hall states at filling fraction nu = 1/k in double-lay
er systems. We found that two power-law anomalies remain in these syst
ems because of the neutral gapless mode. The exponents axe calculated
and we found that the ''orthogonality catastrophe'' exponent is positi
ve. The ''excitonic'' exponent for optical recombination if the electr
on is located in a different layer has the opposite sign. So there exi
st two power-law anomalies for each localized core hole. We suggest th
at the Fermi edge singularity might be measurable in order to detect t
he neutral gapless modes.