Irreducible Witt modules from Weyl modules and $mathfrak{gl}_{n}$-modules

For an irreducible module $P$ over the Weyl algebra $mathcal{K}_n^+$ (resp. $mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $mathfrak{gl}_n$, using Shens monomorphism, we make $Potimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $Potimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $Potimes M$ when it is reducible. Thus we have constructed a large family of irreducible weight modules with many different weight supports and many irreducible non-weight modules over $W_n^+$ and $W_n$.