  We use a complete set of deep narrow-band imaging data for 384 galaxies
gathered during the VESTIGE survey to derive the first Halpha luminosity
function (LF) of the Virgo cluster within R200. The data allow us to cover the
whole dynamic range of the Halpha LF (10^36<LHa<10^42 erg s^-1). After they are
corrected for [NII] contamination and dust attenuation, the data are used to
derive the SFR function in the range 10^-4<SFR<10 Mo yr^-1. These LF are
compared to those derived at other frequencies or using different tracers of
star formation in Virgo, in other nearby and high-z clusters, in the field, and
to those predicted by the IllustrisTNG cosmological hydrodynamical simulations.
The Halpha LF of the Virgo cluster is fairly flat (a=-1.07) in the range
10^38.5<LHa<10^40.5 erg s^-1, and it abruptly decreases at lower luminosities.
When compared to those derived for other nearby clusters and for the field, the
slope and the characteristic luminosity of the Schechter function change as a
function of the dynamical mass of the system, of the temperature of the X-rays
gas, and of the dynamical pressure exerted on the interstellar medium of
galaxies moving at high velocity within the intracluster medium. All these
trends can be explained in a scenario in which the activity of SF is reduced in
massive clusters due to their hydrodynamical interaction with the surrounding
medium, suggesting once again that ram-pressure stripping is the dominant
mechanism affecting galaxy evolution in local clusters of dynamical mass
M200>10^14 Mo. The comparison with the IllustrisTNG cosmological hydrodynamical
simulations shows a more pronounced decrease at the faint end of the
distribution. If Virgo is representative of typical nearby clusters of similar
mass, this difference suggests that the stripping process in simulated galaxies
in these environments is more efficient than observed.
