Hilbert series and obstructions to asymptotic semistability

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants which can be regarded as characters of the Lie algebra of holomorphic vector fields. In this paper we show that, on toric Fano manifolds, the linear span of those Lie algebra characters coincides with the derivatives of the Laurent series of the Hilbert series.