Exponentiation and Fourier transform of tensor modules of $\mathfrak{sl} (n+1)$

With the aid of the exponentiation functor and Fourier transform we introduce a class of modules $T(g,V,S)$ of $\mathfrak{sl} (n+1)$ of mixed tensor type. By varying the polynomial $g$, the $\mathfrak{gl}(n)$-module $V$, and the set $S$, we obtain important classes of weight modules over the Cartan subalgebra $\mathfrak h$ of $\mathfrak{sl} (n+1)$, and modules that are free over $\mathfrak h$. Furthermore, these modules are obtained through explicit presentation of the elements of $\mathfrak{sl} (n+1)$ in terms of differential operators and lead to new tensor coherent families of $\mathfrak{sl} (n+1)$. An isomorphism theorem and simplicity criterion for $T(g,V,S)$ is provided.