  The globular cluster H4, located in the center of the Fornax dwarf spheroidal
galaxy, is crucial for understanding the formation and chemical evolution of
star clusters in low-mass galactic environments. H4 is peculiar because the
cluster is significantly more metal-rich than the galaxy's other clusters, is
located near the galaxy center, and may also be the youngest cluster in the
galaxy. In this study, we present detailed chemical abundances derived from
high-resolution (R~28000) spectroscopy of an isolated H4 member star for
comparison with a sample of 22 nearby Fornax field stars. We find the H4 member
to be depleted in the alpha-elements Si, Ca, and Ti with [Si/Fe]=-0.35+-0.34,
[Ca/Fe]=+0.05+-0.08, and [Ti/Fe]=-0.27+-0.23, resulting in an average
[alpha/Fe]=-0.19+-0.14. If this result is representative of the average cluster
properties, H4 is the only known system with a low [alpha/Fe] ratio and a
moderately low metallicity embedded in an intact birth environment. For the
field stars we find a clear sequence, seen as an early depletion in [alpha/Fe]
at low metallicities, in good agreement with previous measurements. H4 falls on
top of the observed field star [alpha/Fe] sequence and clearly disagrees with
the properties of Milky Way halo stars. We therefore conclude that within a
galaxy, the chemical enrichment of globular clusters may be closely linked to
the enrichment pattern of the field star population. The low [alpha/Fe] ratios
of H4 and similar metallicity field stars in Fornax give evidence that slow
chemical enrichment environments, such as dwarf galaxies, may be the original
hosts of alpha-depleted clusters in the halos of the Milky Way and M31.
