  Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth
manifold $M$, we study the operation, which takes each operator $D$ on the
ambient manifold to a certain operator on the submanifold. The latter operator
is called the trace of $D$. More precisely, we study traces of operators,
associated with actions of compact Lie groups on $M$. We show that traces of
such operators are localized at special submanifolds in $X$ and study the
structure of the traces on these submanifolds.
