Greek character runs house (3)
I believe the answer is:
rho
'greek character' is the definition.
(letter from the Greek alphabet)
'runs house' is the wordplay.
'runs' becomes 'r' (cricket abbreviation).
'house' becomes 'ho' (abbreviation for house).
'r'+'ho'='RHO'
(Other definitions for rho that I've seen before include "letter to Athens" , "Greek letter R" , "Sophocles character" , "Something like 25 [ETA]" , "17th Greek letter" .)