Vanishing resonance and representations of Lie algebras

Abstract

Abstract We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra 𝔤 and the resonance varieties ℛ ( V , K ) ⊂ V * ${\mathcal {R}}(V,K)\subset V^*$ attached to irreducible 𝔤-modules V and submodules K ⊂ V ∧ V $K\subset V\wedge V$ . In the process, we give a precise roots-and-weights criterion insuring the vanishing of these varieties, or, equivalently, the finite-dimensionality as ℂ-vector spaces of certain modules 𝒲 ( V , K ) $\mathcal {W}(V,K)$ over the symmetric algebra on V. In the case when 𝔤 = 𝔰𝔩 2 ( ℂ ) ${\mathfrak {g}}={\mathfrak {sl}}_2(\mathbb {C})$ , 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 ) ${\mathfrak {g}}={\mathfrak {sl}}_n(\mathbb {C})$ or 𝔰𝔭 2 g ( ℂ ) ${\mathfrak {sp}}_{2g}(\mathbb {C})$ , 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.