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With the aid of the exponentiation functor and Fourier transform we introducea class of modules $T(g,V,S)$ of $\mathfrak{sl} (n+1)$ of mixed tensor type. Byvarying 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 explicitpresentation of the elements of $\mathfrak{sl} (n+1)$ in terms of differentialoperators and lead to new tensor coherent families of $\mathfrak{sl} (n+1)$. Anisomorphism theorem and simplicity criterion for $T(g,V,S)$ is provided.

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