We continue the analysis of the dataset of our spectroscopic observation campaign of M31, by deriving simple stellar population properties (age metallicity and alpha-elements overabundance) from the measurement of Lick/IDS absorption line indices. We describe their two-dimensional maps taking into account the dust distribution in M31. 80% of the values of our age measurements are larger than 10 Gyr. The central 100 arcsec of M31 are dominated by the stars of the classical bulge of M31. They are old (11-13 Gyr), metal-rich (as high as [Z/H]~0.35 dex) at the center with a negative gradient outwards and enhanced in alpha-elements ([alpha/Fe]~ 0.28+- 0.01 dex). The bar stands out in the metallicity map, where an almost solar value of [Z/H] (~0.02+-0.01 dex) with no gradient is observed along the bar position angle (55.7 deg) out to 600 arcsec from the center. In contrast, no signature of the bar is seen in the age and [alpha/Fe] maps, that are approximately axisymmetric, delivering a mean age and overabundance for the bar and the boxy-peanut bulge of 10-13 Gyr and 0.25-0.27 dex, respectively. The boxy/peanut-bulge has almost solar metallicity (-0.04+- 0.01 dex). The mass-to-light ratio of the three components is approximately constant at M/LV ~ 4.4-4.7 Msol/Lsol. The disk component at larger distances is made of a mixture of stars, as young as 3-4 Gyr, with solar metallicity and smaller M/LV (~3+-0.1 Msol/Lsol). We propose a two-phase formation scenario for the inner region of M31, where most of the stars of the classical bulge come into place together with a proto-disk, where a bar develops and quickly transforms it into a boxy-peanut bulge. Star formation continues in the bulge region, producing stars younger than 10 Gyr, in particular along the bar, enhancing its metallicity. The disk component appears to build up on longer time-scales.
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