  We compute and study the infrared and sub-mm properties of high redshift ($z
\gtrsim 1$) simulated clusters and proto-clusters. The results of a large set
of hydro-dynamical zoom-in simulations including active galactic nuclei (AGN)
feedback, have been treated with the recently developed radiative transfer code
GRASIL-3D, which accounts for the effect of dust reprocessing in an arbitrary
geometry. Here, we have slightly generalized the code to adapt it to the
present purpose. Then we have post-processed boxes of physical size 2 Mpc
encompassing each of the 24 most massive clusters identified at z=0, at several
redshifts between 0.5 and 3, producing IR and sub-mm mock images of these
regions and SEDs of the radiation coming out from them.
  While this field is in its infancy from the observational point of view,
rapid development is expected in the near future thanks to observations
performed in the far IR and sub-mm bands. Notably, we find that in this
spectral regime our prediction are little affected by the assumption required
by this post-processing, and the emission is mostly powered by star formation
rather than accretion onto super massive black hole (SMBH).
  The comparison with the little observational information currently available,
highlights that the simulated cluster regions never attain the impressive star
formation rates suggested by these observations. This problem becomes more
intriguing taking into account that the brightest cluster galaxies (BCGs) in
the same simulations turn out to be too massive. It seems that the interplay
between the feedback schemes and the star formation model should be revised,
possibly incorporating a positive feedback mode.
