Generalized projective representations for sl(n+1)

It is well known that n-dimensional projective group gives rise to a non-homogeneous representation of the Lie algebra sl(n+1) on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known as Larsson functor), we generalize the above representation of sl(n+1) to a non-homogeneous representation on the tensor space of any finite-dimensional irreducible gl(n)-module with the polynomial space. Moreover, the structure of such a representation is completely determined by employing projection operator techniques and well-known Kostant's characteristic identities for certain matrices with entries in the universal enveloping algebra. In particular, we obtain a new one parameter family of infinite-dimensional irreducible sl(n+1)-modules, which are in general not highest-weight type, for any given finite-dimensional irreducible sl(n)-module. The results could also be used to study the quantum field theory with the projective group as the symmetry.