Nilpotent modules over polynomial rings

Abstract: Let $\mathbb K$ be an algebraically closed field of characteristic zero, $\mathbb K[X]$ the polynomial ring in $n$ variables. The vector space $T_n = \mathbb K[X]$ is a $\mathbb K[X]$-module with the action $x_i \cdot v = v_{x_i}'$ for $v \in T_n$. Every finite dimensional submodule $V$ of $T_n$ is nilpotent, i.e. every polynomial $f \in \mathbb K[X]$ with zero constant term acts nilpotently (by multiplication) on $V.$ We prove that every nilpotent $\mathbb K[X]$-module $V$ of finite dimension over $\mathbb K$ with one dimensional socle can be isomorphically embedded in the module $T_n$. The automorphism groups of the module $T_n$ and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional $\mathbb K[X]$-modules with one dimensional socle.