Simple modules over truncated current Lie algebras of the Witt algebra

Let $\Bbbk$ be an algebraically closed field of characteristic $p>3$, and let $W$ denote the $p$-dimensional Witt algebra, the first example of a non-classical simple Lie algebra. For a non-negative integer $\ell$, consider the associated truncated current Lie algebra $W_\ell=W \otimes \Bbbk[t]/(t^{\ell+1})$. In this paper, we first study simple $W_\ell$-modules having $p$-character $\chi$ of height at most one, and provide a complete classification of such modules up to isomorphism. We then investigate a family of simple $W_\ell$-modules whose $p$-characters have height greater than one.