Irreducible Witt modules from Weyl modules and ${\mathfrak{gl}}_n$-modules

Abstract

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 Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^{+}$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be an irreducible module over $W_n^{+}$ (resp. $W_n$), and determine all submodules of $P\otimes M$ 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 $W_n$.

16 pages