  It is well known that $n$-dimensional projective group gives rise to a
non-homogenous 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-homogenous 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.
