Universal Families on moduli spaces of principal bundles on curves

Abstract

Let $H$ be a connected semisimple linear algebraic group defined over $\mathbb{C}$ and $X$ a compact connected Riemann surface of genus at least three. Let ${\mathcal{M}}_X^{\prime }(H)$ 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 $X\times {\mathcal{M}}_X^{\prime }(H)$ if and only if $H$ is an adjoint group (that is, the centre of $H$ is trivial).