Westerlund 1 (Wd 1) is the most massive stellar cluster in the Galaxy and associated with an extended region of TeV emission. Here we report the results of a search for GeV gamma-ray emission in this region. The analysis is based on ~4.5 years of Fermi-LAT data and reveals significantly extended emission which we model as a Gaussian, resulting in a best-fit sigma of sigma_S = (0.475 +/- 0.05) deg and an offset from Wd 1 of ~1 deg. A partial overlap of the GeV emission with the TeV signal as reported by H.E.S.S. is found. We investigate the spectral and morphological characteristics of the gamma-ray emission and discuss its origin in the context of two distinct scenarios. Acceleration of electrons in a Pulsar Wind Nebula provides a reasonably natural interpretation of the GeV emission, but leaves the TeV emission unexplained. A scenario in which protons are accelerated in or near Wd 1 in supernova explosion(s) and are diffusing away and interacting with molecular material, seems consistent with the observed GeV and TeV emission, but requires a very high energy input in protons, ~10^51 erg, and rather slow diffusion. Observations of Wd 1 with a future gamma-ray detector such as CTA provide a very promising route to fully resolve the origin of the TeV and GeV emission in Wd 1 and provide a deeper understanding of the high-energy (HE) astrophysics of massive stellar clusters.
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