Irreducible modules of $\frak {sl}_{mp}$ in characteristic $p$ with regular or subregular nilpotent $p$-character

Abstract

Let $\bbk$ be an algebraically closed field of prime characteristic $p$. If $p$ does not divide $n$, irreducible modules over $\frak {sl}_n$ for regular and subregular nilpotent representations have already known(see \cite{Jan2} and \cite{Jan3}). In this article, we investigate the question when $p$ divides $n$, and precisely describe simple modules of $\frak {sl}_n$ for regular and subregular nilpotent representations.