A character formula for a family of simple modular representations of $ GL_n $

Abstract.

Let K be an algebraically closed field of finite characteristic p, and let $n\geq 1$ be an integer. In the paper, we give a character formula for all simple rational representations of $GL_{n}(K)$ with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight λ is special if there are integers $i\leq j$ such that $\lambda=\sum_{i\leq k\leq j}a_{k}\,\omega_{k}$ and $\sum_{i\leq k\leq j} a_k\leq {\rm inf}(p-(j-i),p-1)$. Indeed, we compute the character of any simple module whose highest weight λ 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 $GL_{\infty}(K)$-modules.