# A character formula for a family of simple modular representations of $GL_{n}$

### Olivier Mathieu

University of Strasbourg, France### Georges Papadopoulo

University of Basel, Switzerland

## Abstract

Let $K$ be an algebraically closed field of finite characteristic $p$, and let $n≥1$ 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 $i≤j$ such that $λ=∑_{i≤k≤j}a_{k}ω_{k}$ and $∑_{i≤k≤j}a_{k}≤inf(p−(j−i),p−1)$ . Indeed, we compute the character of any simple module whose highest weight $5$ can be written as $λ=λ_{0}+pλ_{1}+...+p_{r}λ_{r}$ with all $λ_{i}$ are special. By stabilization, we get a character formula for a family of irreducible rational $GL_{∞}(K)$ -modules.

## Cite this article

Olivier Mathieu, Georges Papadopoulo, 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