Lefschetz Operators, Hodge–Riemann Forms, and Representations

Abstract

Abstract 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 weight lattice 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 2nd 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.