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

Abstract

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.

18 pages

References in corpus (3)

• Simple sl_{n+1}-module structures on U(h)
• $\mathcal{U}(\mathfrak{h})$-free modules and coherent families
• Classification of simple $W_n$-modules with finite-dimensional weight spaces