Let a reductive group G act on a projective variety +, and suppose given a lift of the action to an ample line bundle θ^. By definition, all G-invariant sections of θ^ vanish on the nonsemistable locus nss+. Taking an appropriate normal derivative defines a map H0(+,θ^)G→H0(Sμ,μ)G, where μ is a G−vector bundle on a G−variety Sμ. We call this the Harder-Narasimhan trace. Applying this to the Geometric Invariant Theory construction of the moduli space of parabolic bundles on a curve, we discover generalisations of “Coulomb-gas representations”, which map conformal blocks to hypergeometric local systems. In this paper we prove the unitarity of the KZ/Hitchin connection (in the genus zero, rank two, case) by proving that the above map lands in a unitary factor of the hypergeometric system. (An ingredient in the proof is a lower bound on the degree of polynomials vanishing on partial diagonals.) This elucidates the work of K. Gawedzki.