Fixed points of Lie group actions on moduli spaces: A tale of two actions

In this paper, we examine Lie group actions on moduli spaces (sets themselves built as quotients by group actions) and their fixed points. We show that when the Lie group is compact and connected, we obtain a linear constraint. This constraint makes the problem of finding fixed points one of representation theory, greatly simplifying the search for such points. We obtain a similar result when the Lie group is one-dimensional. For compact and disconnected Lie groups, we show that we need only additionally check a finite number of points. Finally, we show that the subgroup fixing an equivalence class in the moduli space is a compact Lie subgroup.