Westerlund 1 under the light of Gaia EDR3: Distance, isolation, extent,
  and a hidden population
  (Abridged) We have used Gaia EDR3 data, together with spectra of a large
sample of luminous stars in the field surrounding Westerlund 1, to explore the
extent of the cluster. We carry out a non-parametric analysis of proper motions
and membership determination. We investigate the reddening and proper motions
of several dozen OB stars and red supergiants < 1 deg away from Wd 1. We
identify a population of kinematic members of Wd 1 that largely includes the
known spectroscopic members. From their EDR3 parallaxes, we derive a distance
to the cluster of $4.23^{+0.23}_{-0.21}$kpc. Extinction in this direction
increases by a large amount around 2.8 kpc, due to dark clouds associated to
the Scutum-Crux arm. As a consequence, we hardly see any stars at distances
comparable to that of the cluster. The proper motions of Wd 1, however, are
very similar to those of stars in the field surrounding it, but distinct. We
find a second, astrometrically well-defined population in the foreground
($d\approx2\:$kpc), likely connected to the possible open cluster BH197. Wd 1
is very elongated, an effect not driven by the very heavy extinction to the
East and South. We find a low-density halo extending up to 10' from the cluster
centre, mainly in the NW quadrant. A few OB stars at larger distances from the
cluster, most notably the LBV MN48, share its proper motions, suggesting that
Wd 1 has little or no peculiar motion with respect to the field population of
the Norma arm. However, we are unable to find any red supergiant that could
belong to an extended population related to the cluster, although we observe
several dozen such objects in the foreground. We find a substantial population
of luminous OB members obscured by several more magnitudes of extinction than
most known members. These objects, mostly located in the central region of the
cluster, increase the population of OB supergiants by about 25%.
