The moduli space of symplectic bundles over a compact Riemann surface and quaternionic structures

Let $ G = \text{Sp}(4, \mathbb{C}) $, $ K_G $ be a maximal compact subgroup of $ G $, $ H $ be the subgroup of $ G $ generated by one of the non-trivial elements of the quaternion group, viewed as a subgroup of $ G $, and $ X $ be a compact Riemann surface of genus $ g\geq 2 $. The main result of this paper proves that the forgetful map $ F:M(H)\to M(G) $ between moduli spaces of principal bundles over $ X $ induced by the inclusion $ H\hookrightarrow G $ is a closed embedding. From this, $ M(H) $ can be understood as a closed subvariety of $ M(G) $. Moreover, some applications of this result are provided. In particular, it is proved that the bundles in the image of $ F $ admit a quaternionic structure and also a reduction of the structure group to $ \text{Sp}(2, \mathbb{H}) $. From this, some topological constraints are given, including that the image of the forgetful map falls in a single connected component of $ M(G) $. In addition, some applications are provided concerning the representation space $ \mathcal{R}(\pi_1(X), K_G) $, which, by the Narasimhan-Seshadri-Ramanathan correspondence, is isomorphic to $ M(G) $. Specifically, the image of the forgetful map is proved to correspond to the fixed point subset of a certain subvariety of $ \mathcal{R}(\pi_1(X), K_G) $.