assuming there's no obvious damage to drooping stems, and also assuming your Fatsia has flowered, the flies and likely hoverflies will be there because of the flowers - they like them a lot.

Just check the stems with leaves which are drooping from top to bottom, just to make sure there's no evidence of scale insect, and nothing inhabiting the backs of leaves though. If there's nothing, its likely just the usual response by the leaves to cold weather - they sort of turn flat or downwards, especially if there's been a sudden drop in temperature, or consistent frost.

These shrubs bounce back beautifully after pruning, with lots of bushy growth, so if you think yours is getting too big you could prune it quite hard in Spring once it starts to grow again & thin out any growth that is surplus to your requirements.