Lefschetz operators, Hodge-Riemann forms, and representations

Motivated by a formal similarity between the Hard Lefschetz theorem and the geometric Satake equivalence we study vector spaces that are graded by a weight lattice and are endowed with linear operators in simple root directions. We allow field coefficients in characteristics different from 2. In the case that a “Hodge-Riemann form” exists, the operators (and the grading) yield a semisimple representation of the associated Lie algebra. We then explore the analogous theory with the field replaced by the ring of p-adic integers. In this setup we obtain tilting modules for the associated algebraic group.