The coexistence of localized and free holes in modulation doped InGaAs/InP quantum wells, which show Fermi edge singularities in the cw photoluminescence and excitation spectra, has been determined by picosecond time-resolved luminescence spectroscopy. A long-lived peak at the tow-energy side of the emission spectrum is observed at 5 K and weak excitation. The peak, which vanishes on either increasing the temperature at similar to 30 K or increasing the excitation density, is attributed to recombination of free electrons with localized holes. The simultaneous presence of free and localized holes explains the lineshapes observed in cw measurements and the presence of Fermi edge singularities.