For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple representation of the corresponding Lie algebra if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge-Riemann forms in complex geometry. In the second part of the article we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

中文翻译：

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Lefschetz 算子、Hodge-Riemann 形式和表示

对于特征$\ne 2$ 的域，我们研究向量空间，这些空间由根系统的权重格分级，并在每个简单的根方向上赋予线性算子。我们表明，当且仅当存在满足类似于复杂几何中 Hodge-Riemann 形式的性质的双线性形式时，这些数据扩展到相应李代数的分级半简单表示。在文章的第二部分，我们用 $p$-adic 整数（用 $p\ne 2$）替换域，并表明在这种情况下，某种双线性形式的存在等价于结构的存在关联的简单连接的 $p$-adic Chevalley 群的倾斜模块。