The ‘Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection: genus 0

Abstract

Let a reductive group G act on a projective variety X+, and suppose given a lift of the action to an ample line bundle 0. By definition, all G-invariant sections of 6 vanish on the nonsemistable locus X+ss. Taking an appropriate normal derivative defines a map H°(X+,0)G - ► ^(S^V^)0, where VM is a G- vector bundle on a G- variety *SM. 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.