I am interested in the question: Does there are exist concept of support in representation theory?

When I say support I mean number of non-zero values of $f \in C[G]$. Do you know theorems which talks about the action of elements of $C[G]$ with small support in different representations?

The only example I know about is uncertainty principle which says that for abelian group $supp(f)supp(\hat{f})\geq |A|$.