Lefschetz operators, Hodge-Riemann forms, and representations

TL;DR

This paper explores the conditions under which vector spaces with root system gradings and operators extend to Lie algebra representations, linking bilinear forms to Hodge-Riemann properties and tilting modules over p-adic groups.

Contribution

It establishes a characterization of Lie algebra representations via bilinear forms and connects these forms to tilting modules over p-adic Chevalley groups.

Findings

Existence of a Hodge-Riemann-like form characterizes Lie algebra representations.

In p-adic settings, such forms correspond to tilting module structures.

Provides a bridge between geometric forms and algebraic representations.

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 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.