Arrogant school bowler, perhaps (4,3)
I believe the answer is:
high hat
'arrogant school bowler perhaps' is the definition.
I can't tell whether this definition defines the answer.
'school bowler perhaps' is the wordplay.
'school' becomes 'high' (high is a kind of school).
'bowler perhaps' becomes 'hat' (bowler is a type of hat).
'high'+'hat'='HIGH-HAT'
Can you help me to learn more?
(Other definitions for high hat that I've seen before include "Snobbish, superior person" , "Pair of cymbals on a stand, like lofty headgear" , "snobbish American" , "part of drum kit" , "Snobbish - pair of foot operated cymbals" .)