Vanishing resonance and representations of Lie algebras

TL;DR

This paper investigates the vanishing of resonance varieties associated with Lie algebra modules, providing roots-and-weights criteria and applying these results to classical and geometric contexts, including Green's conjecture and Torelli groups.

Contribution

It introduces a roots-and-weights criterion for resonance variety vanishing and applies it to various Lie algebras and geometric group theory problems, unifying and extending previous results.

Findings

Established criteria for resonance variety vanishing in semisimple Lie algebra modules.

Unified proof of vanishing results for Torelli groups' resonance varieties.

Connected resonance phenomena to classical representation theory and geometric group theory.

Abstract

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the process, we give a precise roots-and-weights criterion insuring the vanishing of these varieties, or, equivalently, the finiteness of certain modules W(V,K) over the symmetric algebra on V. In the case when \g=sl_2(C), our approach sheds new light on the modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves. In the case when \g=sl_n(C) or sp_{2g}(C), our approach yields a unified proof of two vanishing results for the resonance varieties of the (outer) Torelli groups of surface groups, results which arose in recent work by Dimca, Hain, and the authors on homological finiteness in the Johnson filtration of mapping class groups and automorphism groups of free groups.