Universal families on moduli spaces of principal bundles on curves

Abstract

Let H be a connected semisimple linear algebraic group defined over C and X a compact connected Riemann surface of genus at least three. Let M'_X(H) be the moduli space parametrising all topologically trivial stable principal H-bundles over X whose automorphism group coincides with the centre of H. It is a Zariski open dense subset of the moduli space of stable principal H-bundles. We prove that there is a universal principal H-bundle over X × M'_X(H) if and only if H is an adjoint group (i.e., the centre of H is trivial).