A character formula for a family of simple modular representations of $G{L}_n$

Abstract

Let $K$ be an algebraically closed field of finite characteristic $p$, and let $n\ge 1$ be an integer. In the paper, we give a character formula for all simple rational representations of $G{L}_n(K)$ with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight $5$ is special if there are integers $i\le j$ such that $\lambda ={\sum }_{i\le k\le j}{a}_k{\omega }_k$ and ${\sum }_{i\le k\le j}{a}_k\le \inf (p-(j-i),p-1)$ . Indeed, we compute the character of any simple module whose highest weight $5$ can be written as $\lambda ={\lambda }_0+p{\lambda }_1+...+p^r{\lambda }_r$ with all ${\lambda }_i$ are special. By stabilization, we get a character formula for a family of irreducible rational $G{L}_{\infty }(K)$ -modules.