Be someone's pigeon British old-fashioned
In this phrase, the word "pigeon" derives from "pidgin" representing a 17th century Chinese pronunciation of the word `business'.
Be the field in which one is expert; be something which one's responsible for or one has the right to know.
A: Are you the interviewer? B: No. That's John's pigeon-the man sitting over there.
Psychology is not my pigeon. If you want to know more about it, ask John.
Why me? It's Mary's pigeon to collect up the books after class, so she must be responsible for the missing one.
Being looked after or protected by someone
It is used to tell someone that it is time for them to take action or make a decision.
If something falls (squarely) on someone’s shoulders, it is the responsibility that one have to take.
This phrase dates from the 17th century.
To say the same thing repeatedly
Ok, I get it! You sound like a broken record, seriously!