We explore the environmental dependence of star formation timescales in low mass galaxies using the [$alpha$/Fe] abundance ratio as an evolutionary clock. We present integrated [$alpha$/Fe] measurements for 11 low mass ($M_star sim 10^9~M_odot$) early-type galaxies (ETGs) with a large range of cluster-centric distance in the Virgo Cluster. We find a gradient in [$alpha$/Fe], where the galaxies closest to the cluster center (the cD galaxy, M87) have the highest values. This trend is driven by galaxies within a projected radius of 0.4~Mpc (0.26 times the virial radius of Virgo~A), all of which have super-solar [$alpha$/Fe]. Galaxies in this mass range exhibit a large scatter in the [$alpha$/Fe]--$sigma$ diagram, and do not obviously lie on an extension of the relation defined by massive ETGs. In addition, we find a correlation between [$alpha$/Fe] and globular cluster specific frequency ($S_N$), suggesting that low-mass ETGs that formed their stars over a short period of time, were also efficient at forming massive star clusters. The innermost low-mass ETGs in our sample have [$alpha$/Fe] values comparable to that of M87, implying that environment is the controlling factor for star formation timescales in dense regions. These low-mass galaxies could be the surviving counterparts of the objects that have already been accreted into the halo of M87, and may be the link between present-day low-mass galaxies and the old, metal-poor, high-[$alpha$/Fe], high-$S_N$ stellar populations seen in the outer halos of massive ETGs.
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