Olivier Mathieu, Georges Papadopoulo. 
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 
restricted dominant weight $\lambda$ is special if there are
integers $i\leq j$ such that $\lambda=\sum_{i\leq k\leq j}a_k\omega_{k}$ 
and $\sum_{i\leq k\leq j} a_k\leq p-(j-i)$. Indeed, we
compute the character of any simple module $L(\lambda)$ where 
$\lambda=\lambda_{0}+p\lambda_{1}+...+ p^{r}\lambda_{r}$
with all $\lambda_{i}$ are special. By stabilization, we get a 
character formula for a family of irreducible rational
$GL_{\infty}(K)$-modules.