Vanishing resonance and representations of Lie algebras (Q499885)

Let \(V\) be a non-trivial, finite-dimensional complex vector space, and let \(K \subset V \wedge V \) be a subspace. To these data, we can associate two objects: -- The Koszul module, \(\mathfrak W (V; K)\), is a standardly-graded module over the symmetric algebra \(S= \mathrm{Sym}(V) \), given by an explicit presentation involving the third Koszul differential and the inclusion map of \(K\) into \(V \wedge V\) . - The resonance variety, \(\mathfrak R(V; K)\), is a homogeneous subvariety inside the dual vector space \(V^*\), consisting of all elements a \(a\in V^*\) for which there is an element \(b \in V^*\), not proportional to \(a\), such that \(a\wedge b\) belongs to the orthogonal complement \(K^{\bot} \subset V^* \wedge V^*\). These two objects are closely related: at least away from the origin, the resonance variety is the support of the Koszul module. In this paper the authors investigate a relationship between the classical representation theory of a complex, semisimple Lie algebra \(\mathfrak g\) and the resonance varieties \(\mathfrak R(V; K)\subset V^*\) attached to irreducible \(\mathfrak g\)-modules \(V\) and submodules \(K \subset V \wedge V\) . In the process, they give a precise roots-and-weights data insuring the vanishing of these varieties, or, equivalently, the finite dimensionality as \(\mathbf C\)-vector spaces of certain modules \(\mathfrak W (V; K)\) over the symmetric algebra on \(V\) . In the case when \(\mathfrak g = \mathfrak s \mathfrak l_2 (\mathbf C)\). In the case when \(\mathfrak g = \mathfrak s \mathfrak l_n(\mathbf C)\) or \(\mathfrak g = \mathfrak s \mathfrak p_2 (\mathbf C)\), the 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.