I am studying the book "Modern canonical quantum general relativity" written by Thomas Thiemann. In this book, it has been written
since $\vec{N}$ is an asymptotic Killing field of $δ_{ab}$ we have $\mathcal{L}_{\vec{N}}q_{ab} = O(r^{−2})$ odd or $O(r^{−1})$ even respectively for asymptotic translation or boost and rotation while $\delta P^{ab}$ is $O(r^{-2})$ odd.
The only thing I want to know is what the meaning of "$O(r^{-n})$ odd/even" is. This is the first time I encounter this thing and I don't know what it is. Any help would be very much appreciated.
Edit:
Actually I know about odd and even functions but lie derivative of the metric is a tensor, what does even/odd tensor mean?