There are a lot of examples of a group action preserving a specific invariant. For example,
- $\mathrm{SL}(V)$ preserves volume element on $V$
- $\mathrm{SO}(V)$ preserves distance (or more generally, symmetric positive bilinear form) on $V$
- $\mathrm{PGL}(2,\mathbb{C})$ preserves cross ratio
and so on. My question is, is there any standard way to find these invariants when a specific group action is given?