  We consider the space of tensor densities on the n-dimensional sphere with
degree lambda (or, equivalently, of conformal densities with degree lambda).
This space is a module over the group of diffeomorphisms, and consequently over
the Lie algebra of vector fields, on the sphere, and we first prove that as a
module over the conformal group, it is infinitesimally equivalent to an induced
module from the principal series, precisely from the nonunitary spherical
series on the space of smooth functions on the sphere. In function of
lambda,(g,K)-simple and unitary submodules of the space of tensor densities are
therefore classified.
