Coinvariants of Nilpotent Subalgebras of the Virasoro Algebra and Partition Identities

Let V m,n p,q be the irreducible representation of the Virasoro algebra L with central charge cp,q = 1− 6(p − q) /pq and highest weight hm,n = [(np−mq)−(p−q)]/4pq, where p, q > 1 are relatively prime integers, and m, n are integers, such that 0 < m < p, 0 < n < q. For fixed p and q the representations V m,n p,q form the (p, q) minimal model of the Virasoro algebra [1]. For N > 0 let LN be the Lie subalgebra of the Virasoro algebra, generated by Li, i < −N . There is a map from the Virasoro algebra to the Lie algebra of polynomial vector fields on C, which takes Li to z ∂ ∂z , where z is a coordinate. This map identifies LN with the Lie algebra of vector fields on C, which vanish at the origin along with the first N+1 derivatives. The Lie algebra L2N has a family of deformations L(p1, ..., pN+1), which consist of vector fields, vanishing at the points pi ∈ C along with the first derivative. For a Lie algebra g and a g−module M we denote by H(g, M) the space of coinvariants (or 0th homology) of g in M , which is the quotient M/g ·M , where g ·M is the subspace of M , linearly spanned by vectors a · x, a ∈ g, x ∈ M . We will prove the following result.