  In a previous article the second author together with A. Pasquale determined
the spectrum of the $Cos^\lambda$ transform on smooth functions on the
Grassmann manifolds $G_{p,n+1}$. This article extends those results to line
bundles over certain Grassmannians. In particular we define the $Cos^\lambda$
transform on smooth sections of homogeneous line bundles over$G_{p,n+1}$ and
show that it is an intertwining operator between generalized ($\chi$-spherical)
principal series representations induced from a maximal parabolic subgroup of
$\mathrm{SL} (n+1, \mathbb{K})$. Then we use the spectrum generating method to
determine the $K$-spectrum of the $Cos^\lambda$ transform.
