A Conservative in authority shows personality (9)
I believe the answer is:
character
'personality' is the definition.
(I've seen this before)
'a conservative in authority' is the wordplay.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'in' means one lot of letters goes inside another.
'authority' becomes 'charter' (I've seen this in other clues).
'a'+'c'='ac'
'ac' going into 'charter' is 'CHARACTER'.
'shows' acts as a link.
(Other definitions for character that I've seen before include "constitution" , "strength of mind" , "Person in a novel or play" , "Eccentric person" , "Letter; personality" .)