We use Gaia DR2 to measure the initial mass function (IMF) of stars within 250 pc and masses in the range 0.2 < m/Msun < 1.0, separated according to kinematics and metallicity, as determined from Gaia transverse velocity, v_T, and location on the Hertzsprung-Russell diagram (HRD). The predominant thin-disc population (v_T < 40 km/s) has an IMF similar to traditional (e.g. Kroupa 2001}) stellar IMFs, with star numbers per mass interval dN/dm described by a broken power law, m^(-alpha), and index alpha_high=2.03 +0.14/-0.05 above m~0.5, shallowing to alpha_low=1.34 +0.11/-0.22 at m~<0.5. Thick-disc stars (60 km/s < v_T < 150 km/s) and stars belonging to the high-metallicity or red-sequence halo (v_T > 100 km/s or v_T > 200 km/s, and located above the isochrone on the HRD with metallicity [M/H] > -0.6) have a somewhat steeper high-mass slope, alpha_high=2.35 +0.97/-0.19 (and a similar low-mass slope alpha_low=1.14 +0.42/-0.50). Halo stars from the blue sequence, which are characterised by low-metallicity ([M/H] < -0.6), however, have a distinct, bottom-heavy IMF, well-described by a single power law with alpha=1.82 +0.17/-0.14 over most of the mass range probed. The IMF of the low-metallicity halo is reminiscent of the Salpeter-like IMF that has been measured in massive early-type galaxies, a stellar population that, like Milky-Way halo stars, has a high ratio of alpha elements to iron, [alpha/Fe]. Blue-sequence stars are likely the debris from accretion by the Milky Way, ~10 Gyrs ago, of the Gaia-Enceladus dwarf galaxy, or similar events. These results hint at a distinct mode of star formation common to two ancient stellar populations -- elliptical galaxies and galaxies possibly accreted early-on by ours.
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