  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.
