We consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of \( {\mathcal{Z}}_k \)-modules on this space. Here \( {\mathcal{Z}}_k \) is the structure algebra of the root system over a field k. To any wall reection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple rational characters of the connected, semisimple and simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.

中文翻译：

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凹穴和模块表示的交换I

我们将与根系统R相关的一组仿射凹室视为一个拓扑空间，并在该空间上定义\（{\ mathcal {Z}} _ k \）-模轮的某个类别S。其中\（{\ mathcal {Z}} _ k \）是字段k上根系统的结构代数。对于任何墙连接s，我们将S上的墙交叉函子关联。在同伴文章[FL]我们证明了小号编码具有根系统连接，半单和简单地连接代数群的简单合理的字符- [R超过ķ，在该情况下ķ 用高于Coxeter数的特征代数闭合。