Personality of a Conservative in authority (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' (UK political abbreviation).
'in' is an insertion indicator.
'authority' becomes 'charter' (a charter grants authority or authorisation).
'a'+'c'='ac'
'ac' inserted into 'charter' is 'CHARACTER'.
'of' acts as a link.
(Other definitions for character that I've seen before include "strength of mind" , "I, for example" , "Distinctive nature" , "Eccentric person" , "Nature; good reputation" .)