I'm not sure Greek character looking embarrassed should be judged (7)
I believe the answer is:
umpired
'judged' is the definition.
(umpiring is a kind of judging)
'i'm not sure greek character looking embarrassed' is the wordplay.
'i'm not sure' becomes 'um' (expression of hesitation).
'greek character' becomes 'pi' (letter from the Greek alphabet).
'looking embarrassed' becomes 'red' (someone may go red when embarrassed).
'um'+'pi'+'red'='UMPIRED'
'should be' is the link.
(Other definitions for umpired that I've seen before include "Refereed" , "Ripe mud (anag.)" , "Officiated (cricket)" .)