  Let G be a connected reductive group, P its parabolic subgroup. We consider
the parabolic semi-infinite category of sheaves on the affine Grassmanian of G
and construct the parabolic version of the semi-infinite IC-sheaf of each
orbit. We establish some of its properties and relate it to sheaves on the
Drinfeld compactification of the moduli stack Bun_P of P-torsors on a curve. We
also relate the parabolic semi-infinite IC-sheaf with the dual baby Verma
object on the spectral side.
