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

### Athanase Papadopoulos

Université de Strasbourg, France### O. Mathieu

Universität Basel, Switzerland

## 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 5 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 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 $GL_{\infty}(K)$ -modules.

## Cite this article

Athanase Papadopoulos, O. Mathieu, A character formula for a family of simple modular representations of $GL_n$. Comment. Math. Helv. 74 (1999), no. 2, pp. 280–296

DOI 10.1007/S000140050089