We present optical and near-IR data of three Type II supernovae (SNe II), SN 2008bm, SN 2009aj, and SN 2009au. These SNe display the following common characteristics: signs of early interaction of the ejecta with circumstellar material (CSM), blue $B!-!V$ colours, weakness of metal lines, low expansion velocities, and $V$-band absolute magnitudes 2-3 mag brighter than those expected for normal SNe II based on their expansion velocities. Two more SNe reported in the literature (SN 1983K and LSQ13fn) share properties similar to our sample. Analysing this set of five SNe II, which are luminous for their low expansion velocities (LLEV), we find that their properties can be reproduced assuming ejecta-CSM interaction that lasts between 4-11 weeks post explosion. The contribution of this interaction to the radiation field seems to be the dominant component determining the observed weakness of metal lines in the spectra rather than the progenitor metallicity. Based on hydrodynamic simulations, we find that the interaction of the ejecta with a CSM of ~3.6 M$_odot$ can reproduce the light curves and expansion velocities of SN 2009aj. Using data collected by the Chilean Automatic Supernova Search, we estimate an upper limit for the LLEV SNe II fraction to be 2-4 per cent of all normal SNe II. With the current data-set, it is not clear whether the LLEV events are a separated class of SNe II with a different progenitor system, or whether they are the extreme of a continuum mediated by CSM interaction with the rest of the normal SN II population.
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